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THE  INTERPRETATION  O! 
OF  PHYLL01AV 


By 

A.    II 


OXFORI>  *S 

\SGOW 


7/jrrr 


c 

OTANICAL  MEMOIRS.     No.  6 


ON  THE  INTERPRETATION  OF  PHENOMENA 
OF  PHYLLOTAXIS 


CHURCH, 


*HC 


HUMPHREY   MILFORD 

OXFORD   UNIVERSITY   PRESS 

LONDON        EDINBURGH        GLASGOW        NEW  YORK 

TORONTO    MELBOURNE    CAPE  TOWN    BOMBAY 

1920 


CONTENTS 

PAGE 

I.    INTRODUCTORY 3 

II.    GENERAL  PRINCIPLES  OF  PHYLLO-I.-AIS  (Summary)  .       .  6 

III.  ADVANTAGES  OF  THE  EQUIPOTENTIAL  TIIKORY       .       .  16 

IV.  OBJECTIONS  TO  THE  EQUIPOTENTIAL  THEORY         .       .  22 
V.    GENERAL  METHOD  FOR  THE  EXAMINATION  OF  PHYLLO-  27 

TAXIS-PHENOMENA 

VI.    THE  MECHANISM  OF  PHYLLOTAXIS 32 

VII.    PHYLLOTAXIS-PHENOMENA  IN  CRYPTOGAMS  AND  THALLO- 

PHYTA 37 

VIII.    QUINQUELOCULINA 44 

IX.    PHYLETIC  PROGRESSION  IN  PHYLLOTAXIS-MECHANISM      .  50 

LIST  OF  FIGURES     .                                     ,  58 


ON  THE    INTERPRETATION   OF  PHYLLO- 
TAXIS-PHENOMENA 

l 
INTRODUCTORY 

IT  IB  now  some  time  since  a  hypothesis  was  put  forward1  which 
apparently  suggested   reasonable    probability  for  regarding  the 
phenomena   of  rhythmic  patterns  presented  by  ph 
in  the  great  majority  of  higher  plants  as  referable  to 

them 


equal  distribution  of  energy  in  the  growing  plant-apex,  with  the 

the  contact-parastichies  which  map  the  scheme  arc  to  be 

affording  a  clue  to  an  orthogonal  system  of  certain 

in  the  living  substance  of  the  growing-point,  taken  as  a  whole, 

any  regard  to  its  secondary  subdivision  into  component  cells.    The  view 

is  at  first   sight  so  simple,  and  commends  itself  so  readily  to  any  one 

conversant  with   the     eneral  aearance  of  a 


through  a  system  of  growing  leaf-members  at  a 
in  cases  in  which  there  is  little  disturbance  of  the  primary 
that  it  mi-ht  be  thought  that  it  would  have  directly  appealed  to  any 
botanical  observer,  from  the  standpoint  of  the  analogy  of  centric  geometrical 
constructions  for  distribution  of  lines  of  equipotential.  But  it  U  at  once 
obvious  that  no  amount  of  general  resemblance  can  add  anything  to  the 
value  of  the  hypothesis  as  a  scientific  theory  unless  some  proof  can  he 
afforded,  or  suggested,  as  implying  a  reasonable  basis  for  further  progress 
in  the  subject.  The  work  in  Question*  was  not  published  until  inch 
a  suggestion  of  probability  was  forthcoming ;  and  the  initial  difficulty  of 
the  problem,  in  fact,  consists  in  determining  to  what  extent  any  true  proof 
may  be  pos>ible ;  since  angular  measurements  on  actual  plant-toed 
or  on  careful  camera- lucida  drawings  of  sections,  can  never  hope  to 

n  a  range  of  accuracy  admitting  of  an  error  of  less  than  half  a 
while  precise  mathematical  theory  soon  begins  to  tabulate  rotnui 
seconds.     However,  taking  the  mathematical  deduction  of  the  theory  •§ 
involving  a  divergence-angle  of  approximately  1 37 J°  for  all  FT 
ratios  of  (2  : 3)  and  over,  it  is  remarkable  how  nearly  this  angle  If  l 
mated  in  actual  measurements  of  sectional  drawings  taken  from 
presenting  such  phyllotaxis-fonnatioos. 

The  method  of  proof,  admitted  as  the  only  one  possible 
circumstances,  consisted  in  assuming  the  (act  of  orthogonal 

tion  of  unil 


as  presented  in  systems  involving  a  simple  condition 

and  passed  on  to  the  deduction  of  equations  for  the  sectional 

theoretical  primordia  involved  in  such  pi 

tion  of  their  geometrical  properties.   To 

was  given/   and   the  form  of  several  of  then 

lly  interesting  as  differing  to  the  eye  from  the  shape  of  a  true  drde) 


Church  (1901),  Ani^BoC  ir,p.48i,'No*eoo  Pbi-flotaxii':  (1904).  'On 
Mechanical  Uw*':    (1904).  Ana. 


•  **  \  —  y  "i 

Relation  of  Phyllouxts  to  Mechanical  Laws';    (1904).  Ana.  Bou  si*,  p.  117. 
•  Principles  of  PhrUotaxis ' ;  Cook,  (1914)  'The  Qm*t  of  Life '.  p>lf. 

•  Rcl.  Ph.  Mech.  Laws,  ICK  ,50.  »  loc.  OL,  p.  jtf.    Fl*  DL 


4          On  the  Interpretation  of  PJitnorncna  of  Phyllotaxis. 

was  plotted  and  figured  for  reference.1  The  remarkable  fact  was  obtained 
from  the  mathematical  review  of  the  problem,  that  all  such  curves  are 
necessarily  symmetrical  with  regard  to  a  radius  of  the  system^  whether 
the  construction-lines  arc  spiral  or  circular?  From  this  fact  the  conclusion 
was  drawn  that,  since  these  remarkable  properties  were  common  to  the 
quasi-circle  figures  of  the  uniform  growth  construction,  and  also  to  the 
typical  leaf-forms  of  plant-life—in  that  the  quasi-circle  possesses  the  'doi-i- 
ventrality '  as  well  as  the  '  isophylly '  of  a  foliage-leaf  under  all  circumstances 
of  normal  centric  distribution3— one  is  legitimately  entitled  to  assume  that 
the  mathematical  facts  afford  so  strong  an  inherent  probability  of  actual 
agreement,  that  the  hypothesis  is  well-supported,  and  is  entitled  to  further 
extension  along  similar  lines.  So  far,  then,  it  would  appear  that  the  general 
proposition  of  uniform  growth,  involving  primordia  of  the  type  of  the  quasi- 
circlc,  affords  a  firm  foundation  for  obtaining  the  first  fundamental  views 
of  the  mode  of  approaching  the  subject  of  the  mechanism  of  leaf-form 
in  the  plant-kingdom ;  and  this  should  be  capable  of  transference  to  the 
theory  of  the  construction  of  all  lateral  growths  included  in  living  organ 
under  the  term  '  appendages  '. 

Since  problems  of  phyllotaxis  attract  little  attention  at  the  present 
time,  when  there  is  so  much  to  assimilate  in  other  branches  of  the  science, 
and  the  introduction  of  an  excessive  amount  of  mathematical  calculations 
renders  the  subject  distasteful  to  the  general  botanist,  who  possesses 
a  healthy  scepticism  as  to  the  capacity  of  plant-life  for  restricted  mathe- 
matical presentation ;  and  while  the  subject  of  formal  morphology  still 
remains  obscured  under  such  infinite  complications  of  biological  speciali- 
zation, that  many  have  lost  hope  of  any  scientific  presentation  of  the 
fundamental  factors  of  plant-form  as  may  have  been  outlined  by  the  older 
writers  on  morphology ;  and  we,  so  to  speak,  no  longer  see  the  wood  for 
the  trees ;  it  may  be  well  at  the  present  stage  to  summarize  the  points 
in  favour  of  the  present  theory,  and  to  state  concisely  the  difficulties  of  the 
subject  as  well  as  its  advantages  and  suggestions.  As  the  mere  mention 
of  Tangential,  Equiangular,  or  Logarithmic  Spirals,  the  curves  utilized  in 
these  preliminary  constructions,  suffices  to  throw  the  non-mathematical 
botanist  off  the  subject,  the  idea  may  be  conveniently  summed  up  as  the 
Equipotential  Theory  of  Phyllotaxis ;  thus  indicating  in  one  word  the 
necessary  essential  geometry  of  the  constructions,  and  the  suggestion 
of  the  Physiological  Mechanism  of  the  production  of  the  patterns ;  while 
it  also  covers  the  wider  case  in  which  the  construction  may  be  presented 
in  terms  of  straight  lines  and  circles  (whorls),  as  well  as  the  more  complex 
and  changing  systems.4 

The  Equipotential  Theory,  if  it  then  be  so  termed,  in  order  to  avoid 
any  introduction  of  the  word  spiral,  which  has  led  to  so  many  pitfalls  and 
futile  discussions,  makes  no  new  assumptions  whatever,  beyond  features 
regarded  as  fundamental  by  the  recognized  leaders  of  the  past.  Thus 
principles  of  orthogonal  construction  were  first  demonstrated  in  the  plant 
by  Sachs,  though  not  in  this  particular  connexion ;  and  the  existence  of 
orthogonal  series  of  leaf-appendages  was  assumed,  though  not  proved  by 
Schwendener.6  Again,  while  the  full  interpretation  of  all  space-form  is 
necessarily  based  on  the  mathematics  and  geometry  of  three  dimensions, 


1  R.P.M.L.,  p.  336.  *  loc.  cit.,  p.  332. 

*  loc.  cit.,  p.  332.  4  loc.  cit.,  pp.  in,  174- 

*  Sachs(i887),  'Physiologyof  Plants ',Eng. Trans.,  p.  497  ;  Schwendener (1878), 
'Mechanische  Theorie  der  BlaUstellungen ' :   Weisse,  in  Goebel's  <  Organography ' 
(i900,Eng.Trans.)f  p.  74. 


Introductory. 

and  the  elaboration  of  a  plant  or  animal  growth-form  in  the 
organism!  U  still,  from  our  elementary  standoints.  an  infin 
phenomenon,  there  is  no  reason  why  the  first 


phenomenon,  there  is  no  reason  why  the  first  fundamental  principle*  timid 
not  be  firmly  established  ;  and  in  the  application  of  the  Kquipotentiei 
Theory  to  plant-ph>  1  .vc  have  at  hand  a  remarkable  series  of  secial 


cases  of  centric  space-distribution  which  may  afford  a  clue  to  (ar  wide? 
problems.  No  apology  need  be  offered  for  the  introduction  of  a  theory 
which  is  undoubtedly  destined  to  have  a  brilliant  future  as  one  of  the 
most  fundamental  features  of  biological  science.  Just  as  Plant  Itiysiology 
may  be  defined  as  the  study  of  the  controlling  effect  of  Plant-life  on  the  laws 
s  and  Chemistry,  and  modern  physiologists  are  continually  atalhsf^ 
with  wearisome  iteration,  the  obvious  truth,  that  every  plant- 

~mix~»l    «   ••  nl«  ii      I          •   fc^—        **^A 


mechanism  consists  of,  and  works  in  terms  of,  chemical 
the  range  of  vision  ; !  so  the  study  of  Formal  Morphology  may  he  regarded 
as  that  of  the  similar  directive  effect  of  li 

Geometry ;  the  ultimate  expression  of  which  may  again  come  under  the 

'   of   molecular  arrangements  and  groupings,  with  the   laws   which 

mine  them.      1  he  possibility  is  not  wholly  eliminated  that  in 

with  physiological  processes  there  may  be  other  agencies  beyond 

orga  but   at    the    present    time  it   is  difficult    to  get   tx, 

mole  -.  !  this  n-!M  isn    the  centre  of  attack.     That  U  to  say. 

when   the   relations  of  visible  primordta  and  cell  -unit  %   fail   to  give  an 


adequate  explanation  of  phenomena  of  form  and  structure,  the  next  step 
will  be  the  elucidation  of  the  possible  molecular  organization  of  the 
protoplasm  itself.  This  in  turn  may  fail,  in  the  examination  of  vital 

omcna,  and  the  solution  of  the  problems  be  removed 
further  on.     But  at  the  present  day  we  are  still  approaching 
phenomena  and  have  as  yet  no  conception  of  their  limit.     It  was 
standpoint  that  the  equipotcntial  theory  was  originally  put  forward. 

llotaxis  is  removed  beyond  questions  of  meek 
of  primordia  and  cell-segmentation,  the  question  of 
factors  becomes  the  next  point  of  attack  There  it  no  other 

Reference  may  be  also  made  to  other  papers  fk«Hng  with  the 

problems  of  leaf-origin  within  more  recent  Tears ;  the 


perhaps,  a  volume  by  G.  van  Iterson,  jun.  (\qpiY  covering  a  «ide 
abstruse  mathematical  speculation.    The  principles  of  the  log  spiral 
are  recognized  as  essential,  from  the  standpoint  that 
*  similar  figures '  must  be  inevitably  expressed  in  eq 
ratios  of  the  contact-pa rastichtet  are  adopted  as  the 
ing  the  patterns.    Apart  from  mathematical 


remarkable  feature  is  the  departure  from  the  principles  of 
fttruction,  previously  postulated  as  alone  likely  to  throw  agttt 

HfesMtftfcA vtiswi   f\f  i lw*    f\ff*tfw*4**st    ttfwt   sin    •  ss  •••%nj    14   nvA/4t*   tA  S*VMA£M 

;'       "•      '•":'.       :'.'.<:  '     ':._:.  ::-.'.     r.     .. 

A^WU    ,  t         .  •    mmmm*m.A     • •     Mk     .     u_l_Jlll     M*  "       ' 

VSBDW  pnnoratun  is  assumed,  uaseo  on  a  wnouy  sp 

as  a  curve  supposed  to  imitate  the  general  form  of  a  leal 

convincing  reason  being  adduced  as  to  why  the  projection  of  a  honso 

on  a  cone-sur&ce  should  do  so  (both  circles  and  coMsbefagwtttegaftapha** 

apex).     Log  spiral  systems  of  such  similar  lolioids  (p.  161)  are  pot  farwan!  as 


1  Csapck  (1911),  'There  is  nothing  to  indicate  that  the 
1  by  forces  which  are  different  from  chemical  and  physical 


ruled 
nature*. 

*  Van  Iterson  (1907),  Delft,  •  Matbemaiische 
Studien  uber  Blattstellungen ',  pp,  1-331 


6          On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 

these  views  to  Schwendenerian  systems  of  contact-pressures  and  the  origin  of 
lateral  buds  affords  no  clearer  presentation  of  the  subject  than  was  pre\  i 
available ;  but  it  may  be  noted  that  a  valuable  contribution  to  botanical  literature 
was  afforded  in  the  recognition  for  the  first  time  in  a  lx>tanical  work  of  the 
systems  of  organization,  closely  comparable  so  far  only  as  relating  to  geometrical 
spiral  systems  of  similar  figures,  and  similarly  involving  patterns  with  quite 
definite  Fibonacci  relations,  which  occur  among  holozoic  Protista  in  the  sea  (as 
certain  types  of  Foraminifera,  loc.  cit.,  p.  300). 

For  the  continued  supi>ort  of  contact-pressure  theory,  and  complete  mis- 
conception of  what  the  Equipotential  Theory  is  meant  to  do,  cf.  Weisse *  (1903) 
Priiigs.  Jahrb.  ixzix,  p.  416. 

Schoutc  (1913)*  adopts  the  notation  of  the  (m  +  n)  type,  but  follows  Iterson 

and  the  contact-pressure  theory  of  Schwendener,  in  speculations  which  get 

further  away  from  observation  of  the  actual  conditions  at  the  plant-apex,  with 

special  reference  to  a  type  of  curve  termed  a  '  Pseudoconchoid  '. 

The  great  difficulty  of  phyllotaxis  discussions  appears  to  be  to  steer 

clear  of  mathematics  and  take  facts  as  given  by  actual  plant-forms ;  since 

facts  of  observation  may  be  correct  if  the  interpretation  prove  wrong.     In 

looking  over  such  schemes  as  those  of  Schoute  (pp.  268,  270,  277),  <>r 

Iterson  (Taf.  I,  IV,  V,  VI,  X,  XII),  one  can  only  wonder  what  on  earth 

these  things  have  to  do  with  plant-form,  and  the  older  criticism  of  Sachs  as 

to  '  playing  with  figures '  acquires  a  new  significance. 

The  essential  theory  behind  Phyllotaxis  deals  with  the  causes  of 
leaf-origin,  the  meaning  of  what  is  implied  by  a  '  leaf-extension  of  the 
plant-soma,  its  phylogeny  and  secondary  adaptations ;  packing  within 
a  secondary  bud-construction,  secondary  changes  and  irregularities  in  the 
scheme  of  arrangement,  and  the  working  out  of  complex  leaf-form  as 
the  expression  of  phenomena  of  retarded  unilateral  growth,  are  very 
subsidiary  to  the  main  points  at  issue. 


II 
THE   GENERAL   PRINCIPLES   OF    PHYLLOTAXIS1 

BEYOND  the  vague  attraction  of  the  presumed  mystical  and  decorative 
properties  of  spirals  which  appealed  to  early  human  intelligences,4  in  the 
absence  of  any  knowledge  as  to  the  geometry  of  such  constructions  (beyond 

1  Weisse  (1903),  P rings.  Jahrb.,  p.  416. 

1  Schoute  (1913), '  tJber  Pseudokonchoiden  '.    Rec.  de  Trav.  Ne'erland.  x,  p.  153. 

1  Church  (1904),  '  On  the  Relation  of  Phyllotaxis  to  Mechanical  Laws ',  pp.  353, 
112  ftps. 

The  following  short  summary  of  a  previous  memoir,  on  the  general  nature  of 
the  growth-problems  to  be  dealt  wiih  in  the  consideration  of  the  shoot-systems  of 
higher  plants,  may  serve  as  a  convenient  introduction  to  a  subject  which  is  admittedly 
not  only  complex  and  involved,  but  has  been  made  needlessly  so  by  many  more 
mathematical  writers,  and  is  still  little  known  to  botanists  who  have  been  largely 
content  with  the  barest  outline  of  the  simpler  and  more  superficial  phenomena. 

The  examination  of  much  of  the  detail  is  certainly  tedious  to  many  to  whom  a 
pseudo-mathematical  presentation  is  often  a  deterrent,  and  inferences  have  been 
frequently  drawn  from  gratuitously  introduced  and  wholly  empirical  mathematical 
premises,  which  are  little  justified  by  the  facts  observed  in  plant-constructions.  At 
the  same  time  there  can  be  no  doubt  that  the  subject,  sufficiently  interesting  in  itself 
to  be  a  delight  to  the  investigator  of  organic  and  inorganic  space-form,  has  a 
greater  value  as  the  only  available  stepping-stone  to  a  more  comprehensive  know- 


Tkt  Central  Principle  of  PkyUottxii. 

Archimedes),  culminating  in  such  abstractions  as  the  Spiral  Theory  of 
Goethe's  Nature  Philosophy,'  of  which  traces  persist  to • —  -*- 

.  •          /»  •  .        • 


the  first  step  in  the  more  distinctly  botanical  investigation  of  spiral  growth- 

familiar  <f*i*cmmci4  nstsm,  ai 
' 1  spiral ',  in  terms  of  a  helix  winding  on  a  cylinder,  which  has  done 


forms  begins  with  Bonnet4  who  first  distinguished,  though  without  any 

ar  fwtaMKtW  syium.  a*  the 


strictly  geometrical  presentation,  the  familiar 


in  elementary  text-books  to  the  present  day,  and  has  boon 
regarded  as  a  sufficiently  satisfactory  statement  of  the  (acts  of 
leaf-arrangement,  so  far  as  the  non-mathematical  student  is  coocetncd ' . 


all  further  discussion  involving  fantasies,  possibly  pleasing  to  the  martin 
matician,  but  remote  from  any  practical  application  in  phyaiologfcal 
botany,1  and  to  be  relegated  to  cranks  of  minor  ada 
In  the  more  obvious  facts  of  observation  on 
sidcrablc  progress  was  made  in  France  by  • 
the  more  modern  statement  of  the  problems  followed  the  more  popular 
presentation  of  the  subject  by  Schimper11  and  Braun  (1850-55)  of  the 
rising  German  school  "  ;  and  it  is  with  the  introduction  of  Gcnnanbotankal 
books  that  general  conceptions  have  been  inherited  by  modern  writers.1* 
It  is  interesting  to  note  that  Schimper  and  Braun  really  added  little  to  the 
fundamental  conception  of  Bonnet  and  Calandrini  of  s 
previously,  but  merely  elaborated  the  same 


ledge  of  what  has  been  included  under  the  vague  ideas  of  '•  growth  *  and  •  farm  *  by 

„,  r,  i   i    ,  , 

iii»r  j>(i'    '  '.•  1 1-. 

Academic  standpoints,  whether  in  morphology,  snatoc 
a  considerable  part  in  a  science  which  should  be  a  record  of 
and  it  is  often  only  after  wearisome  wanderings  through  such  b  , 
find  the  clue  to  what  really  lies  behind.    A  few  illustrations  expressing  sossc  of  Bt 
more  important  features  have  been  taken  from  the  same  work  (r in.  I-XI). 

A.  Cook  (1903),  'Spirals  in  Nature  and  Art';  (tft^'Thf  CurwjsofUb9, 
p.  266. 

•  Goethe  (1831),  '  Spiraltendcnx  dcr  Vegetation',  p.  194;  Sachs  (1890.  Bag. 
Trans.), 4  History  of  Botany ',  p.  159. 

•  Bonnet    (1754)  dies  sur    I'ttsage  des    scuffles  dans  les   plsntes*. 
pp.  164-188. 

7  PranU and  Vines (1881), 'Ten-book of  Botany ',0.7;  Vmes(iSaj),'&u4ssttr 
Text-book  of  Botany',    p.  17  ;   Stresburger  (1911,  Eng.  Trans.}  'fen-hook  of 


Botany',  p.  41  ;  Bower  (1919), 'The  Living  Plant '  p.  171. 

1  Sachs  (i875,Eng.  Trans.),  'Text-book  of  Botany',  p.  174.  fooMOtt;  (t8t7.Enf. 
Trans.),  'Physiology  of  Plants*,  p.  501. 


'  From  such  purely  academic  and  pseudo-philosophicml  fanes  of  app 
subject  had  attained  s  bad  name,  even  among  botanists;  and  this  was  MX 
in  more  recent  times  by  the  writings  of  SOlwwfaacf  sod  his  opiiBiiuli  Much 


futile  speculation  even  in  modern  papers,  more  especially  m  dhcu«iOii  of  the  ids* 
lions  of  floral  constructions,  ought  have  been  avoided  by  a  dearer  sscofnUouof  rt» 


elementary  laws  of 

Also  Airy  (1874),  I>roc  Roy.  Soc,  udi,  p.  197;   Htuslow  (ifff} 
Linn.  Soc.  ii,  Vol.  I.  p. 

'•  Br*vais(i837),AnruSci.NtLBo(..  pp.  67-71;  CdeO»dole(ii4$>,'TWork 

de  1'anglc  unique  m  Phyllotaxie  '.  Archiv.  des  so.  pbys.  Ct  math,  nui  p.  Iff. 

"  Schimper-Braun  (1835).  'Flora*,  pp.  145.  737. 

*  Sachs,  'History  of  Botany  '.beat.  p.  161:  a  dear  accouat  of  all  oMer  first  is 
given  by  C  de  CsndoOe  (1881.  Geneva),  •  Coosktoauons  sur  r*ude  de  U  Pbyfto- 
ttude',  Archiv.  des  sd.  phjr^et  math._v.  pp.  160.  358. 


l>  Sachs  (1875,  Eng.  Trans.),  'Ten-book  of  Botany*,  pp.  itf-tt-i ; 
dener  (1878,  ^psig^Iechantsche  Theorie  der 

meke  Bet  Mitt,  i,  p.  105 


8          On  the  Interpretation  of  Phenomena  of  Phyllotcu 

the  full  scries  of  the  ratios  of  the  Fibonacci  series  (or  any  other  summation- 
series),1  and  expressing  their  '  fractional  divergences ',  and  mathematical 
schemes  in  a  manner  which  has  proved  the  joy  of  generations  of  elementary 
mathematicians,  much  as  the  Linnaean  System  of  Classification  satisfied 
generations  of  simple-minded  systematists,  and  has  contributed  to  the 
irritation  of  those  looking  for  first  causes  in  plant-morphology.  Thus 
Sachs,1  as  the  leading  text-book  authority  of  the  middle  nineteenth  century, 
dismissed  the  subject  as  *  playing  with  figures',  yet  failed  to  see  that  though 
such  properties  of  numbers  do  express  some  rhythmic  law,  it  is  the  factor 
inducing  such  rhythm  which  is  the  more  fundamental  problem  of  physiology. 
The  attitude  of  the  general  botanical  public  has  been  admirably  stated  by 
Harvey-Gibson*  (1919),  as  he  recalls  'the  miseries  endured  in  endeavours 
to  master  what  was  regarded  as  one  of  the  articles  of  a  botanist's  faitli '. 

Yet  there  can  be  no  doubt  that  behind  such  facts  there  must  be  one 
of  the  most  fundamental  laws  of  living  plasma,4  the  correct  appreciation  of 
which  may  open  up  the  way  to  a  clearer  comprehension  of  what  is  included 
under  the  expressions  growth  and  form ;  since  all  growth  and  all  form 
in  living  organism  must  have  had  some  beginning,  and  plant  shoot-systems 
do  not  come  '  by  Nature ',  ready-made/' 

The  observations  of  Bonnet  and  Bravais,  Schimper  and  Braun,  by  the 
limitations  of  their  age,  were  necessarily  devoted  entirely  to  the  description 
of  effects  noted  on  adult  plant-shoots — with  no  reference  whatever  to  the 
causation  which  might  have  produced  them.  To  our  ideas,  the  subject, 


1  Van  Tieghem(i89i),'Traue'  de  Botanique',  p.  55;  Sachs  (1875),  Text-book, 
p.  181. 

*  Sachs  (1887,  Eng.  Trans.),  '  Physiology  of  Plants',  pp.  497-499. 

*  Harvey-Gibson  (1919),  'Outlines  of  the  History  of  Botany',  p.  98. 

4  Church  (1919),  'The  Building  of  an  Autotrophic  Flagellate',  Bot.  Mem.  i, 
{-.  ii.'  Polarity  and  Surface  Tension '. 

*  A  recent  authoritative,  if  non-botanical,  pronouncement  on   the   subject   by 
D'Arcy  Thompson  ('Growth  and  Form',   1917,  pp.  634-651),  is  also  of  special 
interest  as  setting  up  views  derived  from  ancient  literature  for  the  sake  of  knocking 
them  down  again,  rather  than  for  any  reference  to  work  on  plants  themselves.     A 
pseudo-mathematical  disquisition  by  Tait  (1872),  which  has  nothing  whatever  to  do 
with  the  way  a  plant  is  made,  is  utilized  (p.  645)  to  explain  the  '  numerous  coinci- 
dences '  and  '  mysterious  appearances ',  until  '  we  come  without  more  ado  to  the 
conclusion  that  the  Fibonacci  series  and  its  supposed  usefulness,  and  the  hypothesis 
of  its  introduction  in  plant-structure  through  natural  selection,  are  all  matters  which 
deserve  no  place  in  the  plain  study  of  botanical  phenomena '. 

After  all,  the  ratios  do  occur,  and  must  be  '  useful*  for  something  ;  the  chances 
are  that  their  '  curious  mathematical  properties '  do  afford  a  clue  to  their  meaning ; 
4  mutation  and  natural  selection '  cover  the  only  means  known  to  science  of  deter- 
mining why  one  plant  should  show  them  and  not  another,  as  specific,  generic,  or  even 
family  '  constants '.  It  is  also  the  privilege  of  the  botanist  to  investigate  anything  he 
comes  across  in  the  plant  whether  plain  or  ornamental. 

For  example,  it  is  difficult  to  explain  to  the  non-botanist  that  Phyllotaxis  does 
not  consist  solely  in  speculating  on  the  appearance  of  Pine-cones  or  Sunflower 
capitula ;  the  former  are  of  interest  only  as  a  special  case,  visible  to  the  naked  eye,  of 
the  obscure  constant  (a  Fibonacci  ratio)  controlling  the  growth  and  space-form  of 
every  leafy  shoot  of  every  Pine-tree  ;  and  the  disk  of  the  Sunflower  similarly  visualizes 
a  mode  of  construction  controlling  the  growth  of  the  vegetative  apex,  and  so  far  the 
morphological  organization  of  the  entire  plant.  Other  trees  and  plants  may  show 
other  constants  equally  established,  and  probably  '  useful ',  dominating  the  vegetative 
space-form,  as  the  decussate  symmetrical  system  of  the  Ash,  Sycamore,  Aesculus  and 
Buxus ;  though  some  of  these  (Sycamore,  Aesculus)  may  present  vestigia  of  Fibo- 
nacci ratios  in  their  floral  organization. 


Tkt  Genera/  Printiple*  of  Pkyt/otoxu.  ^ 

like  many  others,  was  thus  approached  from  the  wrong  end;  it  was  even 
established  and  expressed  in  literary  form  in  complete  ignorance  of  what 
the  plant  was  really  doing ;  so  that  not  only  does  the  entire  superstructure 
rest  on  no  adequate  foundation,  but  its  more  essential  problems  hrroms 
obscured  and  muddled  for  any  subsequent  analysis.1 

Modern  Botany  has  little  to  do  with  the  effect*  which  appeal  to  the  eye 
on  an  adult  plant-shoot.  It  seeks  to  determine  how  these  ptinnnmosji 
originatcd-what  is  the  mechanism  of  their  production,  what  factors  He 
behind  the  mechanism,  and  how  it  was  originally  called  into  operation ; 
that  is  to  say,  for  wh.it  original  function,  or  by  what  response  to  condition* 
of  external  \\  dealing  with  *uch  a  subject  in  ihc  most 

elementary  man:  thus  necessary  to  take  an  unbiased  and  wholly 

fresh  start,  and  begin  it  again  from  a  different  point  of  view. 

Leaf-arrangement  is  a  function  of  the  'growing-point'  of  a  'stem'; 
phyllotaxis  is  the  problem,  not  of  the  final  appearance  of  the  leaves  on 
a  shoot,  but  of  the  origin  of  such  lateral  appendages  at  the  plant-apex— the 
discussion  of  the  causes  which  lead  to  the  inception  of  a  prfmordfam  before 
it  becomes  visible  as  the  slightest  protuberance  of  the  cell-tissue  of  an 
The  subsequent  fate  of  the  primordium,  or  its  behaviour  und 
pressures  is  a  wholly  irrelevant  and  secondary  detail  to  be 
later.  For  example— the  attainment  of  the  adult  condition  in 
which  become  equal  in  volume,  or  are  equally  spaced  by  second 
nodal  extension,  may  lead  to  htlital  effects  on  the  adult  axes,  which  are 
wholly  secondary.  Similarly  all  interpretation  of  such  effects  in  the 
transverse  dimension,  as  plane  figures,  may  be  expressed  in  terms  of  Spirmlt 
of  Arckimtdts,  of  equal  and  uniform  screw-thread,  which  clearly  have  no 

c-ncc  whatever  to  the  initial  factors  of  a  growing  system  of  unequal 
units.  That  the  whole  geometrical  and  mathematical  conception  of 
phyllotaxis-rclations  should  have  been  originally  expressed  in  terms  of 

trnedcan  notation,  was  the  necessary  limitation  of  CaUndrini,  BravaK 

mper  and  Braun,  who  knew  no  other  mathematics  of  spiral 
and  the  subject  once  started  on  these  lines  followed  on  to 
without  it  occurring  to  anybody  to  point  out  such  a 

Once  it  is  understood  that  phyllotaxis  involves 
at  a  growing  and  expanding  apex,  it  is  necessary  to 
of  mathematical  growth,  as  studied,  for  example,  in  the  growth  of  the 
electrical  field,  and  see  to  what  such  generalizations  may  lead.  The  general 
mathematical  proposition  of  a  field  of  uniform  growth  about  a  point  (centric), 
as  represented  by  a  circular  mesh  work  of  •quasi-square*',  is  at 
illuminative ;  since  it  not  only  gives  certain  effects  of  spatial 
in  tcrm.s  of  straight  lines  and  circles,  but  it  provides  for  the 

duction  of  '  growing '  logarithmic  (or  equiangular)  spirals,  »  *  system 
which  is  undergoing  uniform  expansion,  and  to  which  the  phyllouxis- 
phcnomcna  of  a  growing  apex  must  be  obviously  referable  in  the  Jot 
instance,  however  much  such  a  simple  mathematical  postulate  may  prove 
incapable  of  expressing  all  the  facts  of  a  living  organism.  Plant-form  b 
not  a  simple  subject,  to  be  readily  attacked  by  the  expert  mathematician; 
certain  form-relations  may  appear  at  first  sight  simple,  and  have  so  appealed 
to  many  botanists  since  the  time  of  Nehemtah  Grew  (l68t)a;  just  as  any 
child  with  a  compass  may  draw  a  'whorl  of  6  leaves  in  a  drde ' ;  or  even 

1  SchwendeneTlTsTS),  'Mechanise!*  Theoric';  Weis*  (ifoo).  in  GocbsTs 
•Orgsnographjr*,  Eng.  Trrnn^  i,  p.  7$. 

*  Church  (1901),  Ann.  Hot.  xv,  p.  481. 

•  Grew  (1681).  *The  Anatomy  of  Plan**',  p.  i$*.    •  So  frost  ts*  L  j  1 1 1  ssjiii 
tion  of  Plants,  men  might  first  be  invited  to  Msihcsssiifsl  EMMETS':  Tab.  4  J  < 


io        On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 

further  back,  as  in  speculations  by  Kepler1  on  the  occurrence  of  penta- 
merous  symmetry  in  flowers,  and  observations  on  spirals  by  Leonardo  da 
Vinci.  In  dealing  with  biological  phenomena,  the  most  elaborate  human 
mathematics  still  fail  hopelessly  in  touching  such  problems;  yet  the  initial 
steps  must  be  simple  and  readily  followed;  complexities  come  later,  and 
require  to  be  taken  step  by  step  as  in  the  Newtonian  laws  of  motion. 

Comparison  of  such  a  scheme  of  uniform  growth-expansion  about 
a  point  (centric  and  two-dimensional),  with  consequent  possibilities  of 
orthogonal  construction-lines,  as  expressing  lines  of  equipotential  (as  in 
a  static  electrical  field),  or  of  lines  of  equal  pressure  and  flow  in  circular 
or  spiral  vortices  of  fluid  motion,2  is  also  illuminating  as  expressing  a  con- 
venient geometrical  method  of  *  transformation '.  Thus  by  usini;  such 
a  framework  of  '  squares '  it  is  readily  possible  to  transform  any  circular 
scheme  into  a  spiral  '  homologue ',  or  vice  versa ;  and  again  by  added  com- 
plexity to  express  the  corresponding  eccentric  homologues  of  either  centric 
;>iral  systems.3  But  such  mere  mechanism  of  transformation,  though 
immediately  interesting  as  exhibiting  the  true  botanical  relation,  so  long 
misunderstood,  of  the  essential  geometrical  relation  of  spiral  and  whorled 
phyllotaxis.  so  widely  distributed  in  the  plant-kingdom,  is  but  a  small  part 
of  the  range  of  the  subject.  The  essential  significance  of  the  construction 
of  such  a  field  of  growth,  is  that  it  \&  growing  \  i.e.  it  expresses  a  mccha 
in  continuous  operation,  expanding  according  to  definite  laws,  and  adding 
new  units  from  the  central  point.  That  this  feature  is  often  curiously 
missed  or  miscomprehended,  has  been  well  illustrated  by  D'Arcy  Thompson 4 
(1918),  who  in  discussing  phyllotaxis-phenomena  as  presented  by  Cook,6 
naively  states  that  he  does  not  see  anything  mysterious  in  the  mechanism 
of  phyllotaxis,  as  hinted  by  the  latter  writer,  beyond  simple  pattern-building. 
The  most  interesting  section  of  this  work  on  'Growth  and  Form'  is 
devoted  to  the  exhibition  of  schemes  of  transformation  as  applicable  to 
animal  growth-forms ;  and  it  is  clear  that  to  a  zoologist  conversant  with 
the  growth-increase  of  an  animal  to  one  adult  form  once,  the  conception 
of  a  'growing-point',  and  its  wholly  mysterious  property  of  continuing 
to  build  similar  new  forms  indefinitely,  is  wholly  unfamiliar  and  unrealized 
owing  to  the  limitation  of  the  static  diagram. 

The  initial  step  in  any  future  consideration  of  Plant-growth,  as  leading 
to  the  initiation  of  leaf-members  at  a  growing-point  is  thus  the  utilization 
of  the  geometrical  proposition  of  uniform  centric  growth  about  a  point ; 
the  system  being  considered  in  two-dimensional  form,  as  in  the  transverse 
plane  of  a  botanical  section  ;  since  the  longitudinal  component  only  adds 
the  factor  of  linear  extension  of  the  system,  and  such  growth-spirals  cannot 
be  studied  in  three  dimensions  ;  the  effect  of  the  theoretical  dome-shaped 
apex  of  the  plant-shoot  being  again  the  expression  of  secondary  phenomena 
of  growth-retardation.  That  no  finite  plant-body  can  be  satisfactorily 
imitated  by  a  mathematical  construction  expressed  in  terms  of  extension 
to  infinity,  is  sufficiently  obvious ;  but  finite  propositions  may  also  come 
later  on.  The  simplest  mathematical  conception  has  to  be  taken  first,  as 
satisfying  some  of  the  primary  features  of  the  problem,  but  not  all :  the  finite 
stage  of  the  proposition  may  be  reached  in  the  introduction  of  retardation- 

1  Kepler  (1611),  cf.  Ludwig  (1896),  '  Weiieres  tiber  Fibonacci-Curven ',  Hot. 
Centralbl.  68,  p.  7. 

|  Tail,  Enc.  Brit.,  ix  Edit.,  vol.  15,  p.  723  (Mechanics). 

5  Church  ('904).  Ann.  Bot.  xviii,  p.  227.     Fig.  I. 

4  D'Arcy  Thompson  (1918),  '  Growth  and  Form ',  p.  639. 

1  T.  A.  Cook,  'The  Curves  of  Life',  p.  81,  'The  Meaning  of  Spiral  Leaf- 
arrangement  '. 


The  Central  Primcipfo  of  Pkylfataxis.  \  \ 


effects,  which  in  plant-constructions  are  atsociifrd  with  the 

of  an  adult-form,  on  the  ultimate  complete  cessation  of  further  growth  ;  the 

growing  system  being  thus  brought  to  rest* 

Leaving  on  one  side  all  traditional  obsessions  to  favour  of 
cylindrical  shoot-systems  of  adult  axes ;  i 
of  the  system,  and  taking  the  two-dimensional  express 
section  of  a  growing-point,  it  is  soon  realized  that  in 
the  Icaf-primordia  follow  a  perfectly  straightforward 
ment,  in  all  normal  growing  centric  systems— expressed  to  terms  of 
curv  ing  and  intersecting  in  opposite  senses,  the  number  and  ratio 

of  which  may  be  readily  checked  and  scheduled;  e.g.  in  s 
(JN  +  »)  or  (m:*).  The  latter,  as  including  the  property  of 
intersecting  log  spirals,  may  be  utilized  from  such  a  preliminary  i 
though  open  to  the  objection  that  the  curves  seen  can  never  be  log 
on  any  growing  plant  in  which  the  growth-rate  b  never  "^ttmtH 
uniform.  All  phyllotaxis-constructions  are  thus  scheduled  to  terms  of 
intersecting  curves  (normal  phyllotaxis);  such  regularity  following  as  a 
consequence  of  rhythmic  production,  i.e.  uniform  worktop  to  the  machine. 
if  there  is  no  •  pattern  '  there  is  no  rhythm,  and  conversely.  The  fact  that 
many  phyllotaxis-constructions  in  the  plant  are  so  broken,  tod 
hopeless  of  interpretation,  b  thus  legitimately  regarded  as  the 
of  failure  in  the  mechanism  to  retain  co-ordination ;  and  b  to  be 
biologically  as  implying  decadence  of  the  construction-system;  as.  for 
example,  in  plant-formations  in  which  such  systems  may  be  uestigisl,  and 
now  useless  for  any  practical  purpose,  and  hence  the  more  luff  rest  tog  to 
their  phylogenetic  significance.  Such  cases  abound,  as,  for  egampte.  to 
floral  construction  (cf.  androccium  of  Poppy,  the  sporophyUs  of  a  large 
Clematis,  the  pattern  of  the  achcncs  on  a  strawberry),  or  the  great  bud  of 
Eqnuftum  Telmatfia  with  vestigial  and  rudimentary  leaf-teeth, 

But  however  interesting  it  may  be  to  schedule  such  numerical  (actors 
and  their  secondary  complications,  the  interpretation  of  phyllotaxb  b 
concerned  with  something  more ;  not  so  much  the  mere  (acts  of  the  rhythsm, 
as  the  causes,  still  obscure,  to  which  phenomena  of  rhythm  are  the  response. 
Thus,  in  the  first  place  .simple  generalizations  follow  front  the 

geometrical  relations  of  such  curve-patterns;  and  these  apply  to  all  pattens 
expressed  as  lines  distributed  around  a  point,  whatever  the  nature  of  the 
«  .  g. : — 

(a)  When  m  =  n  the  construction  b  symmctruml.  that  b  to  say.  ex- 
pressed in  a  complementary  system  of  straight  lines  and  circles,  (Fig.  IV  j 

(ft)  If  m  and  «  are  unequal,  the  construction  becomes  'Atjmm****' 
or  spiral,  as  the  most  general  mathematical  case,  and  b  so  far  to  be 
regarded  as  '  primitive'  in  the  biological  sense,  to  absence  of  any  reason 
to  the  contrary.  Fig.  II.) 

(y)  In  any  such  asymmetrical  construction,  lines  expressed  by  sun 
(i»  +  n)  and  difference  (*-*)  of  the  primary  numbers  will  map  out  a 
plemcntary  system,  as  diagonals  of  the  meanes  of  the  original 
and  other  Systems,  as  expressed  by  continued  *tara  of '  sum 
will  pass  through  the  same  points,  with  others.    The  occurrc 
tion-scries  of  numbers  expressing  ratios,  has  no  special  relation  to  d 
but  is  the  expression  of  certain  general  pc opcrties  of  such  systenw  of 
secting  curves. 

(J)  If  m  and  *  have  unity  only  as 
I  through  all  the  points  of  intersection. 


Church  (1904).  Ann.  Bou  *&.  p.  si*. 


12         On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 

(c)  Where  *H  and  //  have  a  common  factor  (e.  g.  2,  3),  no  one  spiral 
will  include  all  the  points;  but  as  many  spirals  will  pass  through  that  part 
of  them  as  arc  indicated  by  the  common  factor  (e.g.  two  spirals  each 
through  half). 

(()  Where  m  and  n  only  differ  by  unity,  the  grand  spiral  passing 
through  all  the  points  is  conspicuous  as  the  diagonal  complementary 
system. 

(B)  The  points  of  intersection,  or  areas  delimited  by  any  such  curve- 
•cysti  from  any  unit,  arc  readily  given  a  numerical  value,  or 

•  numbered  up',  by  the  obvious  properties  of;//  and  //  distribution  ;  a  simple 
device  (Braun,  1835)  which  has  always  proved  irresistible  to  botanical 
writers,  though  adding  nothing  to  the  original  factors  of  the  case,  and 
usually  proving  very  confusing  to  the  beginner,  who  is  apt  to  take  the 
numbers  as  unalterable.  (Fig.  XII .1 

It  remains  to  consider  to  what  extent  plant-constructions  bear  out 
generalizations;  that  is  to  say,  supply  examples  of  all  the  possible 
mathematical  cases  which  may  arise  ;  it  being  so  far  clear  that  there  is 
no  special  virtue  or  mystery  about  a  spiral  curve,  except  that  it  may  not  be 
so  easy  to  draw  as  a  circle  with  a  compass ;  all  log  spirals  are  in  fact  curves 
intermediate  between  the  straight  line  and  the  circle;  or,  preferably,  the 
general  expression  of  a  growth-movement  of  which  the  straight  line  and 
the  circle  are  the  limiting  cases.  Since  an  absolutely  straight  line,  or 
a  true  circle,  is  inconceivable  in  a  living  plasmatic  organism,  it  may  be  said 
that  all  curves  in  organic  nature  are  based  on  logarithmic  spirals ;  though 
again  never  absolutely  attaining  to  the  accuracy  of  such  mathematical 
conceptions,  and  liable  to  a  wide  range  of  secondary  changes. 

On  comparing  the  data  given  by  plant-apices,  it  is  very  evident  that  all 
the  general  phenomena  of  curve-systems  are  abundantly  exemplified,  and 
common  examples  should  be  familiar  to  every  botanical  student.  For 
example : — 

(a)  The  symmetrical  construction  is  that  long  known  in  botany  as 
'  whorled '  phyllotaxis,  in  which  members  of  successive  whorls  alternate.1 
Superposed  symmetrical  constructions  are  exceedingly  rare,  and  do  not 
obtain  in  any  case  in  which  they  can  be  regarded  as  primitive.  All 
symmetrical  systems  are  again  suggestively  derivative  from  earlier  conditions 
of  '  spiral '  organization ;  the  alternation  of  the  whorls  being  in  fact  the 
strongest  evidence  in  this  connexion. 

(/3)  The  case  of  asymmetrical  phyllotaxis,  in  which  the  numbers  have 
no  common  factor,  is  distinctly  the  most  general  case ;  the  original 
recognition  of  spiral  leaf-arrangement  is  based  on  this  fact.2  (Figs.  VI,  VII.) 

(y)  Complementary  systems  are  classed  as  'pa rastichies '  of  various 
grades  ('  orthostichus '  also  coming  under  this  rule),  commonly  confusing 
the  analysis  of  the  pattern  on  the  part  of  earlier  writers  (cf.  'multiple 
spirals '  of  Bonnet) :  the  primary  curves  of  the  system  may  be  distinguished 
as  *  contact-parastichics  '.3 

(8)  The  single  grand  spiral  is  in  fact  the  feature  commonly  appealing 
to  observers  of  elongated  axes,  from  the  times  of  Bonnet  and  Schimper; 
so  much  so  that  it  has  been  widely  accepted  as  a  causal  factor  (' genetic 
spiral') ;  whether  rightly  or  wrongly  is  still  a  point  at  issue. 

(()  The  case  of  the  common  factor,  early  recognized  by  Bravais  as  the 


1  R  P.M.L.,  loc.  cit,  p.  142,  'The  Symmetrical  Concentrated  Type'. 
*  Leonardo  da  Vinci,  xvi  cent.,  Bonnet,  xviii  cent. 

3  '  Sliding-growih  effects '  may  render  the  contact-parastichies  obscure  in  older 
constructions ;  cp.  R.P.M.L.,  loc.  cit.,  pp.  315,  345. 


The  General  Prifuiffa  of  Pkyllotaxi*. 


Hi  jugate,  Trijugate.  &c.,  construction—  a*  a  sort  of 
giving  both  whorls  and  spirals  at  the  tame  time,  and  hence  a 
may  be  sporadic  in  any  line  (cC  Pine-cone,  HffanuJbu  capftulttm).  but 
appears  characteristic  in  Diptacu*  and  several  /ty*sr*ss  (c£  also  Sdf4t*mi. 
hat  no  single  'genetic  spiral  '  can  be  postulated  for  such  a  construction. 
it  gives  the  f**t  at  grfa  to  all  theories  of  the  causal  lytrwfHi  of  such 


a  single  construction  line,  and  its  divergence-angles,  in  the  case  of  the 
Flowering  Plant.1 

if)  Given  the  interesting  case  of  •  spire 'construction ;  the  grand  spiral 
(m-m)  being  complementary  to  (m  +  n)  more  nearly  vertical  series,  and 
hence  presenting  as  many  obvious  'spires'  as  are  expressed  by  the  sum  of 
the  construction-curves ;  also  sporadic  in  many  series  (ct  S*/*m,  Cjt 
Euphorbia  tif/atu/ubsa).  .uui  allowing  from  the  .  4  the  Fibonacci 
scries  in  5-spircd  floral  constructions  imitating  superposed  whorls,* 

(6)  Where  higher  ratios  obtain,  the  method  of  numbering  up  will 
bring  out  the  contact-relations  in  the  simplest  manner ;  while  failure  to  give 
connected  results  in  this  respect  is  the  criterion  of  the  irregular  system  and 
deteriorated  mechanism ;  the  latter  often  local  and  expressed  in  varying 
degree  of  inaccuracy. 

To  this  may  be  added  as  facts  of  observation :-  - 

(i)  The  enormous  preponderance  of  Fibonacci  number*  in  spiral 
systems,  to  the  extent  that  any  other  ratio  may  be  termed  relatively 
rare  and  exceptional,  with  the  deduction  well-warranted  that  Fibonacci 
phyllotaxis  may  be  regarded  as  phylogenetically  primitive  ;  while  all 


constructions  are  variants,  up-grade  (symmetrical)  or  down-grade  (irregular). 
(a)  The  fact  that  Fibonacci  numbers  are  lost  in  secondary  w hoi  led 
constructions  with  high  numbers,  though  wonderfully  persistent  as  WKribftsstf 
( i  +  i ),  dfcussatf  (a  +  2),  trimtrous  (3  +  3),  symmetry,  and  curiously  vestigial 
as  in  the  5  of  pentamerous  flowers  of  Dicotyledons,  and  the  'qutncuoaal* 
caly 

(3)  In  no  case  is  one  number  of  the  ratio  more  than  twice  the 
(m :  a  m)*  the  range  being  between  m :  m  (wheeled)  and  ». 

these  cases  are  again  relatively  so  few  as  to  be  regarded  as  an 
undoubtedly  secondary  as  '  sports '  and  '  mutants '  of  the  original 

(4)  Owing  to  the  general  effect  of  the  growth -pressure*  of 
in  close  lateral  contact,  the  theoretical  points  of  intersection  are 
replaced  by  more  or  less  rhomboidal  areas,  bounded 

to  which  all  the  same  generalizations  apply.    (Fig.  VI  > 

Given  such  data  it  should  be  possible  to  recai 
1  divergence-angles  ',*  in  terms  of  log  spiral  constructions,  to 
ccptions  of  the  value  of  bulk-ratio,1  or  the  fact  that  different 
same  plant  may  produce  similar  leaf-members  in  different 
correlation  with  the  relative  volumes  of  axis  and  appendage,  and  it  becomes 
apparent  that  older  generalizations  of  the  Schtmpcr-Braun  type  have 
touched  but  the  fringe  of  a  very  remarkable  subject ;  while  any  tom^us* 
methods  involving  the  acceptance  of  older  Archimedean  notations,  to  he 
'corrected*  in  the  process  of  initiation,  or  subsequently  " 
'displacements'  of  primordia,  on  the  lines  indicated  by 
again  express  a  regrettable  ignorance  of  the  essential 
problems. 

1  R.P.M.L..  •  Muliijagaie  Types',  p.  166. 

•  loc.  i .:..    Ix*st  Concentrated  Type '.  p 

»  toe.  at.  p.  341.  •  toe.  ciu  p.  J4* 

•  kx.  cit.  p.  338. 


14        On  the  Interpretation  of  Phenomena  of  Phyllotaxi<. 

Undoubtedly  the  most  important  point  brought  out  by  the  examination 
of  shoot-apices  in  which  primordia  make  their  first  appearance,  is  ih< 
that  every  form  of  phyllotaxis-construction  which  is  rhythmic,  and  works 
out  a  pattern,  is  readily  expressed  in  terms  of  such  intersecting  curves,  and 
in  no  other  manner  (cf.  the  bijugate  system,  and  the  relation  of  whorls  to 
spirals).  This  applies  more  convincingly  to  the  case  of '  rising-phyllot. 
of  the  type  of  Hclianthus  leafy  stem  and  inflorescence ;  the  rules  deduced 
for  the  latter  also  explaining  the  contact-relations  of  the  beautiful  bijugate 
expansion-system  of  the  /?//.r<ir//,r-head,  and  similarly  applicable  to  more 
irregular  and  decadent  'falling'  constructions.  In  all  such  cases,  though  it 
may  be  possible  to  trace  the  single  '  genetic  spiral '  by  the  method  of 
numbering  the  units,  one  can  never  be  wholly  convinced  that  a  complex 
construction  has  been  adjusted  and  so  elegantly  fitted  together,  cither  by 
minute  changes  in  '  divergence-angles ',  or  by  obscure  mechanism  of  mutual 
pressure  of  certain  units  in  the  scheme,  so  much  that  the  system  must 
express  the  simultaneous  growth  of  a  curve-complex,  expanding  or  con- 
tracting according  to  some  definite  law.1  (Fig.  VIII.) 

Similarly  all  local  variations  in  the  original  system  are  readily  followed 
in  terms  of  the  addition  or  loss  of  the  primary  parastichy-curves,  one  at 
a  time,  with  consequent  alteration  of  the  pattern  in  other  respects.  For 
example,  a  (6:7)  construction  (cf.  Cactaceae),  by  adding  either  a  '  lon<j 
curve',  or  by  losing  a  '  short  one',  would  attain  true  circular  symmetry  with 
whorls  as  (7  :  7)  and  (6 :  6)  respectively :  or  again  by  adding  a  '  short  curve ' 
it  would  appear  as  (6 : 8),  a  bijugate  form  :  in  all  such  cases  the  original 
'  genetic  spiral  '-effect  would  vanish.  Conversely  the  addition  of  curves  one 
at  a  time  to  a  symmetrical  system,  as  in  Casuarina  and  Equisetum^  would 
give  a  spiral  *  staircase  '-effect ;  though  in  these  highly  specialized  types  the 
occurrence  is  so  rare  as  to  pass  as  a  '  freak '. 

Excessive  variations  of  this  type  soon  lead  to  hopeless  irregularity ; 
and  local  irregularities  wholly  destroy  any  rhythmic  effect.  In  extreme 
cases  one  side  of  a  plant  may  appear  normally  constructed,  and  the  other  be 
quite  irregular  and  indeterminate.2 

Ample  evidence  is  thus  at  hand  that  phyllotaxis-mechanism  of  the 
shoot-apex  is  to  be  referred  to  an  initial  choice  of  curve-systems,  comparable 
with  the  isolation  of  equal-spaced  lines  of  growth-potential ;  and,  so  far  as 
the  Flowering  Plant  is  concerned,  is  distinctly  not  referable  to  any  single 
'genetic  spiral',  working  out  as  a  causal  factor  the  spacing  of  members 
at  some  specially  thought  out  divergence-angle,  with  the  inconceivable 
accuracy  of  even  seconds  of  arc  in  apices  commonly  only  a  fraction  of 
a  millimetre  in  diameter.  The  principle  of  the  log  spiral  construction 
based  on  a  view  of  uniform  growth-expansion,  in  the  first  instance,  merely 
introduces  the  conception  of  growth  at  some  uniform  rate,  and  is  the 
mathematical  expression  of  a  centric  field  of  growth  involving  the  expansion 
of  the  units  as  '  similar  figures '.  Any  set  of  similar  figures,  whatever  their 
shape  will  necessarily  fall  into  log  spiral  lines ;  and  it  has  been  already 
noted  that  in  plant-coastructions  the  '  similar  members '  are  not  absolutely 
similar,  since  they  are  subject  to  secondary  growth-changes,  and  the 
'  log- spirals '  are  never  absolute.  But  assuming  a  leaf-origin  to  imply 
a  subsidiary  and  equally  centric  disturbance  of  the  primary  centric  field  of 
growth,  it  may  be  legitimate  to  regard  such  primordia  as  essentially 
ISO-diametric^  and  in  such  case  they  must  bear  some  relation  to  simpler 
iso-diametric  geometrical  units  as  '  squares '  or  circles '.  It  is  again 


1  R.P.M.L.,  p.  in,  174. 

*  loc.  cit.,  p.  97  :  anomalous  head  of  Dipsacus,  p.  173. 


The  General  Principle*  of  Pkjilotaxu.  1 5 

difficult   to  distinguish  which   may  be   the  causal  factor  and  which  list 
consequence. 

On  the  other  hand,  taking  such  an  interpretation  of  the  jrnulh  eiiissi 
sion  scheme  as  following  such  equally  spaced  lines  of  orthogonal  intersection. 
as  most  readily  drawn  quasi -square*,  it  is  easy  to  pass  on  to  the  rrt^*pOtm 
of  the  quad-circle;  just  as  with  even  greater  facility  the  idea  of  the  latter  if 


obtained  by  merely  postulating  that  every  lateral  primorJutm  it 

a  centnc  A 


from  a  joint  as  a  centric  disturbance  in  a  centnc  growA  sj**m  The 
remarkable  geometrical  attributes  of  the  atusi-drcle,  as  localiaad  in  aa 
orthogonally  arranged  system  of  such  units,  have  been  formulated  by 
Hayes,1  and  the  curve  itself  is  defined  in  such  a  manner  that  the  initial 
form  of  the  primordium,  referable  to  any  given  ratio,  as  considoad  la 
the  same  two-dimensional  transverse  section,  can  be  plotted  as  a  stai 

reference;  as  in  the  expression  (log  j)'**  »  ^jT^jf  (*%•  IX>» 


prejudice  to  the  fact  that  such  curves  will  not  obtain  in  any  plant  in 
nature  ;  though  the  approximation  may  be  cloter  in  the  higher  ratios. 
and  the  essentials  exist  in  all,  in  a  secondarily  modified  form.  The  special 


features  of  these  remarkable  curves  are  of  the  general  character  noticed 
in  bud-sections,  and  closer  approximation  may  be  readfly  ffttpprftfltij  by 
adding  the  effect  of  a  secondary  gro»  th-reurdation  in  one  or  both  dimen- 
sions.1 This  fact  may  be  left  for  future  consideration:  but  so  far  the 
geometrical  construction  and  mathematical  properties  of  fyiisi  rirrks 
undoubtedly  do  afford  an  illuminative  explanation  of  all  the  cvential  and 
primary  attributes  of  a  leaf-appendage  accepted  by  an  older  school  of 
academic  morphologists,  contemplating  the  plant  for  over  a  century,  la 
complete  ignorance  of  its  phyletic  origin  in  submarine  environment,  aad 
so  far  analyzed  in  text-books  as  constituting  the  (i)  Bilaterally.  («)  Dorm 
vcntrality,  ($  I  soph  ylly,  of  the  typical  leaf-lamina.  That  all  leaf  primordia 
necessarily  present  these  essential  attributes  follows  directly  from  the 
postulated  orthogonal  log  spiral  construction,  and  not  from  any  other 
construction.  The  last  point  of  bilateral  symmetry  with  regard  to  a  radius. 
in  a  system  otherwise  wholly  spiral  and  obliquely  asymmetrical,  is  in  fact. 
the  detail  which  adds  the  coping-stone  to  the  superstructure  of  log  spiral 
theory.8 

That  the  reduction  of  Phyllotaxis-phcnomcm 
vegetation  to  a  problem  of  orthogonally  intersecting 
mapping  out  points  of  origin  for  new  centres  of  lateral  growth. 
subject  still  an  unsolved  problem,  is  sufficiently  obvious ;  but  the  fact  remains 
that  no  other  conclusion  appears  possible.  No  causal  factor  has  been 
outlined,  and  in  such  case  it  can  be  only  conceded  that  the  coustruction- 
schcmc  may  be  possibly  the  expression  of  an  inherited  mechanism,  or  the 
adaptation  of  some  such  scheme  of  growth-distribution  which  has  been 

ited  in  some  previous  stage  of  aquatic  existence;  as  a 
than  the  vegetation  of  the  land,  to  be  traced  back  to 
horizon  of  plant-life ;  though  now  adapted  and  improved  to  suit  the  • 
complex  conditions  of  subaerial  existence,  in  further  correlation  with 
increased  specialization  of  the  shoot-system  of  'axes'  and  'app 
That  is  to  say,  one  must  be  prepared  to  admit  that 
may  be  the  more  modern  adaptation  of  some  older 
possibly  now  working  on  lines  only  remotely  com 
and    that,  as    in   other  biological  'adaptations', 

•  R  JMI.U,  p.  330.  33*-  '  tec,  csu  r 

•  loccit,  pp.33i.  333- 


1 6        On  the  Interpretation  of  Plicnomena  of  Phyltota.\ 

mechanism,  and  even  working-unit^,  may  have  been  something  quite 
different  from  what  might  be  concluded  from  the  study  of  land-vegetation 
alone.  There  can  be  no  doubt  that  abstract  conceptions  of  what  a  plant 
ought  to  be,  the  expression  in  text-books  of  what  generations  of  academic 
morphologists  have  evolved  from  their  inner  consciousness  and  the  con- 
templation of  land -vegetation  alone,  have  taken  very  much  for  granted. 


Ill 

ADVANTAGES  OF  THE  EQUIPOTENTIAL  THEORY 

I.  As  points  in  favour  of  the  equipotcntial  theory  may  be  not 

(i)  It  replaces  the  Archimedean  notation  of  Schimper  and  Brann, 
well-known  in  the  form  of  '  divergence-fractions ',  clearly  incapable  of 
expressing  the  construction  of  a  growing  apex  (and  never  intended  to  do 
so),  by  a  construction  in  terms  of  growing  spirals.  That  is  to  say,  it 
replaces  an  obviously  false  notation  by  a  more  correct  one.  The  notation 
employed  merely  expresses  the  simple  facts  of  observation,  about  which 
there  can  be  no  dispute,  in  a  form  which  is  also  mathematically  innocuous ; 
i.e.  so  many  paths  cross  so  many,  of  the  form  (m  +  n).  The  use  of  a  ratio 
(m:n)  or  (*»/«),  which  would  definitely  imply  a  system  of  orthogonally 
intersecting  log  spirals  should  be  avoided,  since  the  intersections  actually 
observed  on  a  plant  are  not  orthogonal,  and,  as  already  pointed  out,  can 
never  be  regarded  as  such.  The  crossing  is  a  fact  of  observation  ;  the  ratio 
is  a  matter  of  mathematical  theory.  At  the  same  time,  it  is  not  to  be 
denied  that  ratio-formulae  will  be  probably  utilized  in  the  future. 

(ii)  The  principles  of  orthogonal  construction  lead  directly  to  the 
enunciation  of  the  quasi-circle  as  the  fundamental  plane-representation 
of  all  lateral  primordia,  which  involve  circular  fields  of  growth  contained 
within  a  parent  circular  field  ;  while  corresponding  systems  of  quasi-squarcs 
may  afford  convenient  representations  of  primordia  in  close  lateral  contact. 
Such  systems  constitute  a  valuable  Standard  of  Reference  for  dealing  with 
the  properties  and  forms  of  such  primordia,  however  little  they  may  be 
like  them  at  first  sight.  The  increasing  resemblance  of  such  figures  to  the 
transverse  sections  of  leaf-members,  when  secondarily  modified  by  packing 
and  growth-retardation,  will  be  considered  later. 

(iii)  The  quasi-circlc  hypothesis  in  turn  elucidates  the  remarkable 
properties  of  leaf-members,  their  bilaterality,  dorsiventrality^  and  also  the 
essential  isophylly  (otherwise  inexplicable)  of  the  lamina  in  spiral  as  well  as 
in  whorlcd  constructions,  as  a  purely  mathematical  deduction  ;  but  it  at 
once  renders  other  points  clear,  and  in  a  manner  which  was  not  previously 
considered.  In  the  last  case  it  emphasizes  a  remarkable  property  of  leaf- 
form,  so  accepted  as  a  commonplace,  that  it  had  never  attracted  any  special 
notice ;  though  attention  has  been  paid  to  the  interpretation  (usually  teleo- 
logical)  of  anisophylly. 

(iv)  It  again  renders  great  service  in  clearing  up  the  difficulty  of  the 
'  slipping '  of  the  members  of  spiral  systems  which  formed  so  prominent 
a  feature  in  Schwcndener's '  Contact -Pressure  Theory '.  Since  the  primordia 
of  a  spiral  system  are  bilaterally  symmetrical  with  regard  to  a  radius  of  the 
whole  system  so  long  as  they  are  free  from  one  another,  while  under 
contact- pressure  they  tend  to  be  squeezed  into  the  asymmetrical  meshes 
of  a  packed  spiral  pattern ;  the  '  slip '  of  the  free  portion  of  the  member 


Advantage*  of  tkt  EqmpotemJiat  Tktory.  ,  : 

as  it  escapes  its  neighbours  thus  tends  to  make  it  recover  its  original  radially 
symmetrical  relations  with  regard  to  the  main  axis ;  so  that,  in  the  Uo*. 
its  tangtnHal  diameter  becomes  again  a  circular  path  of  the  main  ijslaav 
instead  of  a  spiral  path  in  the  packed  system.  This  simple  solution  of  the 
Schwendcncrian  •  slip ',  which  may  be  plotted  geometrically,  at  once  pots 
the  contact-pressure  theory  out  of  court ,  quite  apart  from  the  (act  that 
Schwendencr,  in  postulating  circular  primordia  in  orthogonal  series  had 
laid  the  foundation  of  the  cquipotential  theory,  if  he  had  not  been  lost  In 
helical  constructions* 

The  most  interesting  application  of  the  theory  undoubtedly  lies 
in  the  manner  in  which  it  amplifies  the  conclusions  of  Wiesner  on  the 
principle  of  the  maximum  illumination  of  the  members  of  a  leafy 
As  shown  by  Wiesner,  many  years  ago,  the  most  plausible  interpret 
of  the  occurrence  of  numbers  belonging  to  the  Fibonacci-scries  in 

•dated  with 


kingdom,  is  that  they  are  associated  with  the  geometry 

and  involve  a  divergence-angle  which  will  promote  m 

of  the  overlapping  series  of  members.  The  standpoint  was  not  entirely 
since  1  •  he  pioneer  of  spiral  construction,  gave  as  the  '  final 

cause*  of  spiral  phyllotaxis  the  biological  standpoint  that '  TriMtp*  *tim 
h  takfs  place  in  the  leaves  demands  that  air  ikctM  circnlale  /reefy 
*d  them,  and  that  tkty  should  overlap  as  littU  as  pouibU  \  thus  giving 


from  the  first  the  conception  of  the  advantage  of  minimum  supcrposMoo. 
was  obviously  in  accordance  with  the  physiological  ideas  of  the  time, 
when  the  nutrition  of  plants  was  considered  as  effected  solely  by  the 
absorption  of  substances  from  the  soil  by  means  of  the  roots.  After  Sachs 
had  successfully  demonstrated  the  essentials  of  photosynthesis  (i860,  it 
became  possible  for  Wiesner  to  restate  the  problem  in  terms  of  light  (1875). 
Hut  standpoints  still  remain  intermingled,  when  it  is 

that  exposure  to  light,  by  promoting  excessive  transpiration  (c 
tion),  may  soon  reach  a  point  at  which  the  emission  of 
overpass  the  limit  of  the  root -supply,  and  xerophytic 
Thus  maximum  superposition  may  be  advantageous  at  one  period,  bat 
directly  injuriou  !u T,  and  the  plant  has  to  effect  some  sort  of 

promise.     Hence  phyllo:  itions  arc  intimately  associated  with 

ditions  of  both  maximum  and  minimum  superposition;  it  is  a  a 
keep  one's  attention  too  fixed  on  the  former  standpoint  alone. 
ued  that  for  all  scries  of  phyllotax is- fractions,  the  one 
Fibonacci  numbers  gave  the  most  equal  distribution  on  the  axis  for  a 
number  of  leaves.  That  equal  balancing  of  the  leafy  shoot,  which  to 
admirably  attained  by  all  whorlcd  systems,  is  not  the  essential  feature,  waa 
also  claimed  by  Wiesner  (1902);'  and  the  mathematical  deduction  " 

th.it  for  a  given  number  of  leaves  a  divergence  of  3         5  of  560* 

give  minimum  superposition,  while  }  of  360°  gives  the  maximum.  These 
deductions  of  Wiesner  were  purely  mathematical,  and  were  based  on  the 
familiar  series  of  fractional  divergences  of  Schimpcr — } . } ,  | ,  | ,  * 

I 

expressed  as  values  of  the  continued  fraction  To  suit  that 


x  was  taken  as 

1  Wiesner  ^ 875),  Flora,  58,  p.  113.  ' 
tionalc  Divergcnien  * ;   (1901),   Benchtc.  to,   p.  84, 
Stellungsverhaltnisse  der  Laubbllttcr  tor  ~ 

VI  B 


of  560*  -  aaa*  ao'  fT 


i,  and  the  limiting  angle 


1 8        On  the  Interpretation  of  Phenomena  of  Phylto taxis. 

The  result  would  be  written  more  conveniently  as  the  inverse  angle,  with 
x  =  2,  since  the  ratios  J,  f,  f ,  f ,  &c.,  are  complementary  to  the  preceding, 

3  —  J  *\ 
in  which  case  the  limiting  angle  =  -         -  of  360°  =   137°  30'  28-936" 

(Bravais),  the  familiar  '  Ideal  Angle*  of  Schimper.  This  angle,  which  has 
so  long  persisted  as  a  somewhat  mystical  conception,  has  been  always 
repugnant  to  a  number  of  botanists,  to  whom  the  '  progression  towards 
perfection'  implied  by  a  rise  in  the  phyllotaxis-fraction  appeared  wholly 
unreasonable,  from  the  standpoint  that  the  majority  of  leafy  shoots  are 
in  low  divergences ;  and  there  is  no  evidence  whatever  that  the 


highest  divergence  known  (cf.  disk-florets  of  Hclianthus,  as  compared  with 
foliage-leaves  of  the  stem)  has  anything  to  do  with  a  more  perfect  perception 
of  illumination.  In  his  later  paper  (1903)  Wiesner  still  remains  handicapped 
by  this  retention  of  the  helical  constructions  of  Schimper  and  Braun,  which 
again  were  mathematical  expressions,  and  not  simple  data  obtained  from 
the  plant  itself.  For  example,  it  is  abundantly  clear  that  if  the  divergence 
angle  of  137°  30'  28-936"  is  the  Ideal  Angle  for  maximum  illumination, 
the  plant-shoot  which  is  so  commonly  classed  as  a  '}'  construction,  with 
a  divergence-angle  of  144°,  is  so  far  removed  from  the  optimum,  that  one 
hesitates  to  see  the  particular  advantage  of  the  mechanism  which  can  allow 
such  a  constant  error  in  the  angle. 

Once  the  subject  is  freed  from  the  obsessions  of  these  helical  divergence- 
angles,  which  have  no  connexion  with  a  growing-apex,  it  is  interesting  to 
compare  the  results  given  by  the  assumption  of  orthogonal  construction. 

Thus  the  systems  become : — 

I.  On  adult  shoot  with  equal  inter  nodes.          II.  At  the  growing-point. 

|  of  360°  =  144°.  (a  +  3)  =  138°  27'  42"- 

|  of  360°  =  135°.  (3  +  5)='3703«'5o". 

&  of  360°=  138°  2/41-54".  (5  +  8)  =  137°  3''  41"- 

These  figures  suffice  to  show  that  the  optimum  angle  may  be  actually 
attained  within  one  degree  for  the  (2  +  3)  system ;  while  at  (5  +  8)  (cf.  Finns, 
Aspidium^  Hclianthus)  it  may  be  attained  within  almost  one  minute  ;  if,  that 
is  to  say,  the  mechanism  the  plant  has  at  its  disposal  for  building  the 
system  be  sufficiently  accurate.  Apart  from  mathematical  accuracy,  it  may 
be  said  that  the  plant-constructions  indicated  by  these  formulae  may  really 
attain  the  postulated  angle  with  an  accuracy  sufficient  for  all  practical 
purposes,  or  what  is  equally  important,  as  near  as  we  can  measure  it, 
as  well  in  the  lower  systems  as  in  the  higher.  Such  a  conclusion  again 
brings  Wiesner's  generalization  immediately  within  practical  range.  There 
can  be  no  doubt  as  to  the  correctness  of  his  standpoint ;  but  the  log  spiral 
theory  is  needed  before  the  argument  can  be  regarded  as  finally  established  ; 

and  it  can  so  far  be  definitely  stated  that  the  angle  2     _j5  of  360°,  the 

*  Ideal  Angle ',  is  no  longer  a  mystical  conception  of  aim  on  the  part  of 
the  plant,  but,  within  the  range  of  one  degree,  an  actual  mathematical 
property  of  Fibonacci  phyllotaxis  for  all  ratios  of  (2+3)  and  upwards. 
The  fact  remains  that  Wiesner  formulated  these  principles  with  imperfect 
mathematical  data  ;  with  corrected  figures  they  are  plain  to  any  one  ;  at  any 
rate  so  far  as  the  significance  of  the  Fibonacci  numerals  in  plant-construction 
is  concerned. 

From  the  fact  that  Fibonacci  phyllotaxis  gives  optimum  illumination 
to  a  vertically  adjusted  leafy  shoot,  it  follows  that  any  deviation  from 
Fibonacci  systems,  either  as  '  anomalous '  or  whorled  constructions,  implies 


Advantage  of  tk*  E<tuipot**H*l  T*»ry.  ,0 


a  definite  diminution  of  exposure,1  which  may  be  utilised  as  a 
adaptation';  and  as  a  matter  of  fact  such  constructions  are.  on  the  whole, 
charactc  xcroroorphic  vegetation,  though  by  no  means  invariably 

so.  On  the  other  hand  it  is  equally  important  to  icnuimhti  that  direct 
formation  of  a  more  or  less  overlapping  pattern  is  bet  one  means  oet 
of  many  by  which  the  plant  is  able  to  control  its  light-supply,  la  interned*' 
extension,  petiole-formation,  leaf-dissection,  diminution  of  surface,  erection, 
drooping,  petiolar  torsion,  &c,  &c,  we  have  to  do  with  a  whole  aeries  of 


mechanisms  which  correct  or  adjust  the  exposure  of  the  leaf-lamina,  wtat- 
may  be  the  initial  pattern  as  built  at  the  growing-point  of  the  shoot. 
A  pattern,  the  construction-factors  of  which  may  be  hereditary  and  settled 
at  a  relatively  distant  point  in  the  phytogeny  of  the  race,  may  i  si  si  IS 
subsequent  compensatory  adjustments ;  that  is  to  say.  the  mutation  which 
marks  a  new  and  successful  adjustment  does  not  necessarily  involve  the 
initial  pattern  at  all,  but  may  be  something  wholly  fHftrss*  A  chaafa 
of  pattern  represents  only  one  possibility  out  of  many,  and  these  probably 
in  the  long  run  even  more  effective.  Plants  with  perfect  Ffhoeerri  ayateaaa 
may  be  secondarily  adapted  for  extreme  xerophytk  rtmdWfmf  (cf. 
of  Sempervnmm,  reduced  needle- leaves  of  A  rout  aria  tg</Ua).  jest  as 
with  systems  presenting  minimum  exposure  may  be  secondarily  adji 
for  diffuse  light.  Similarly,  while  the  vegetative  pan  of  a  plant  may  present 
a  system  with  considerable  overlapping  (decussate  :  +  i),  the  floral  axis 
involving  purely  non-assimilatory  members  may  revert  to  the 
Fibonacci  pattern  (cf.  Calyca*tk*st  foliage  (a  +  a),  flower  ($  +  8) 
In  fact  the  paradox  remains,  so  manifold  and  so 


compensations  of  the  initial  scheme,  that  the  most  perfect 

purely  'mechanical'   phyllotaxis-patterns  only  survive   in 

which  do   not    include   leaf-members  devoted  to   photosyi 

Hence  the  scales  of  the  Pine-cone,  the 

Composites,  the  spines  of  leafless  Cacti,  and  the 

afford  at  the  present  day  the  most  classical  ea 

which  arc  inherited  and  accurately  followed; 

useless  from  their  primary  significance,  they  retain  their 

as  equally  balanced  constructions.     Similarly  the  best- 

a  change  of  the  phyllotaxis-pattern  under  the  direct  effect  of 

is  to  be  seen  in  a  plant  which  has  no  leaves  at  all  (Crrrnr). 

So  involved  are  the  general  phenomena  of  leaf-arrant 
common  examples  may  be  utilized  to  exemplify  the  fact  that  Wiesner*s  Law 
would  be  of  phy logcnctic  rather  than  of  ootogenetic  significance 
relation  of  the  system  to  the  water-problem  may  be  as  vital  as 
incnts  of  light  for  photosynthesis. 

Thus  (i )  the  common  Ivy  (H*4r*\  which  protest  aooag  sad 

in  normal  Fibonacci  (2  +  3)  series,  to  the  coodttton  for 


Desjiftduced(i^i)ruimmgthcKMsinthauiiislie€tioafar 
(this  being  constant  for  tome  varieties,  e.g.  met  shoots  of  H  **JT,  %*f .  *rfer«a). 
The  (2  +  3)  condition  is  evidently  the  priori**  form,  aad  to  Msodastd  w*  *a 
smeroos  floral  comtroctioo ;  bat  the  mutation  to(l+l)!seoi 


pHirtSIHfMQUf 

a  success :  when  growing  against  a  surface,  with 

terns  are  almost  equally  unactable  sad  reqdrtiODt  uoey  aasaii  i  by 

production  of  internodts,  (a)  petiokr  toraon. 

its  origin  to  a  ledDctioo-proceaa  invortieg  fewar  leaves  sea 

/_\    Ifnliai^   •tinrrfa  fj     ^---*-rmm   mm^  mlft   am  Mftf  Bfi  frn^B    t^Bfltt  CBM   BSCB>  W9 

\2f    roll«gC-HIOOU  ul    fmfmtfm  frwrm»9  •§«  ^nn 


1  Wiesncr  (1901),  lotciL,  p.  (97);  ( 
Biolog.  CentralhL,  23.  fx  209. 

•    : 


2O       On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 

The  leaves  are  small,  well-spaced  apart  on  relatively  long  internodes,  and  all 
appear  equally  suitable  for  photosynthesis,  so  far  as  can  be  judged  by  the  eye  ; 
yet  one  is  bound  to  assume  that  the  less  frequent  (2  +  3)  Fibonacci  system  is  the 
primitive  one  for  the  species,  not  merely  from  the  fact  that  one  may  be  biased 
in  favour  of  the  system,  but  that  the  letramerous  flowers  of  Fuchsia  are  undoubt- 
edly reduced  from  corresponding  pentamcrous  ones;  and  these  in  turn  HUM 
have  had  the  familiar  quincuncial  calyx  which  is  the  Fibonacci  construction.  It 
is  further  clear  that  while  (2  -I-  2)  is  now  the  normal  type  for  the  assimilatory 
shoots,  the  systems  (34-3)  and  (4  +  4)  occur  as  amplifications,  due  to  increased 
vigour  in  shoot-production  ;  the  method  of  superposition  is  not  altered,  but  the 
number  of  leaves  is  increased,  giving  greater  activity  to  transpiration  and  photo- 
synthesis. 

(3)  The  well-known  case  of  seedlings  of  Cmus  and  Phyllocaclus  affords 
examples  of  such  constructions  as  (2  +  2)=  4-ridged,(2  +  3)=  s-ridged,  and 
{3  +  3)  =  6-ridged  plants,  the  ridges  being  developed  along  the  orthostichy  lines. 
On  exposure  to  strong  light  the  cladodes  change  to  3-ridged  =(1  +  2)  and  2-ridged 
=  (i  +  i )  systems.1     The  phyllotaxis  construction  of  these  leafless  plants  is  thus 
modified  to  suit  the  production  of  3-  or  2-angled  cladodes.     Such  a  case  of 
direct  modification  is  almost  inconceivable  in  the  case  of  a  leafy  shoot  in  which 
compensatory  adjustments  may  be  more  readily  effected  in  the  case  of  members 
and  structures  already  in  existence.     In  a  leafless  Cactus  the  only  alternative  is 
a  simple  cladode-flattening  which  need  not  involve  the  phyllotaxis-system  at  all ; 
and  this  occurs  noticeably  in  Opuntia,  the  original  phyllotaxis  pattern  straggling 
over  the  flattened  laminae.     Nor  can  the  case  of  Phyllocactus  be  regarded  as 
solely  due  to  the  direct  effect  of  light.     Strong  plants  of  Ccreus,  pruned  hard 
back  to  the  bare  stems,  similarly  send  up  an  abundance  of  new  young  shoots  in 
the  same  construction-systems,  and  these  again,  as  they  grow  old,  change  into 
3-  and  2-ridged  laminae  in  just  the  same  way,  when  growing  all  the  time  under 
constant  conditions  of  illumination.     It  is  increasingly  evident  that  the  water- 
supply  is  the  essential  factor,  and  the  apparent  effect  of  light  is  due  to  the  greatly 
enhanced  chloro-evaporation.     In  such  cases  the  phenomena  are  of  the  nature 
of  reduction-phenomena  as  xerophytic  adaptations  in  a  starved  apex.*     It  is 
much  easier  to  understand  the  direct  effect  of  want  of  water,  or  diminished 
supply,  in  the  growing  apex,  than  to  postulate  a  mechanism  for  light-perception 
in  the  assimilatory  regions  which  may  be  conducted  to  the  growing-centre  at 
which  the  pattern  is  being  initiated.     In  these  shoots  also  it  is  evident  that  the 
(2  +  3)  system,  which  is  associated  with  the  general  tendency  to  retain  Fibonacci 
systems  in  the  construction  of  the  floral  perianth,  is  to  be  regarded  as  the  phylo- 
genetically  older  system;    in  which  case   the  (24-2)  and  the  (3  +  3)  shoots 
appear  as  the  nearest  simple  variants  on  the  pattern,  both  equally  likely  to  occur 
in  the  case  of  a  plant  to  which  the  mechanism  of  minimum  superposition  is  no 
longer  essential. 

Equally  interesting  in  this  connexion  is  the  case  of  Eucalyptus,  the  lUue 
Gum:  with  the  supersession  of  the  decussating  blue  'juvenile  leaves'  by  the 
more  xerophytic  drooping  'scimitar  leaves',  the  ancestral  (2  +  3)  Fibonacci 
system  is  either  immediately  or  very  soon  regained  :  this  may  be  taken  as  indi- 
cating that  the  decussate  arrangement  of  the  juvenile  leaves  is  really  a  relic  of  an 
older  adaptation  in  the  special  condition  of  the  seedling. 

(4)  Lastly,  when  expressed  in  low  ratios,  it  is  interesting  to  note  how  small 
an  alteration  (in  the  form  of  a  '  mutation ')  may  change  the  phyllotaxis-system 
from  a  position  of  maximum  to  that  of  practically  minimum  exposure.     Thus 
the  (2  -f  2)  system  affords,  on  the  whole,  the  most  general  example  of  a  super- 
posed system  utilized  by  xerophytic  plants ;  yet  the  mere  addition  of  one  new 


1  V6chting(i894),  Prings.  Jahrb.,  xxvi,  p.  438. 

s  Molisch  (1912),  Sitzungsberichte  k.  Akad.  Wien,  p.  833,  gives  a  similar  case  in 
which  three  hours'  exposure  to  radium  emanation  induced  a  deteriorated  effect  in 
apices  of  Sedum  Sicboldii,  a  (3  +  3)  construction  'reducing'  to  (2  +  2),  as  usual  in. 
starved  shoots. 


Advantages  of  the  Ef*ifiotoUM  Tk*tj. 

path  renders  it  (s  +  3)  «*"  til  the  advsntsfss  of  the  Ffcoaacd  isna 
another  new  path,  by  changing  it  tc 
superposition.    All  three  examples 


another  new  path,  by  changing  it  to  (3  4-  3>  ffiws  agate 

oocw  to  thtlovtrt 


again  in  the  inflorescences  of  A  ^wMmm,  in  which  cats*  ibry  CM  h««r  M> 
•pedal  significance  from  the  standpoint  of  Hgta,  but  aw  SOTT?  •niiliifcj  of 
construction;  useful  00)7  to  point  the  moral  of  the  nsrttom  wiih  vt**  **•>- 


essential  phyUocazis^constants  nay 

simple  numerical  factors  may  be  alone  variable  in  a 


So  far,  then,  Wiesner's  Law  is  the  best  ideological  repression  of  the 

and  for  maximum  illumination,  and  may  be  provisionally  regarded  as 

-  the  'aim  '  of  thr  l.tml -plant  under  favourable  conditions  of  Isfht 

and  water-supply.     Hut  maximum  illumination  may  soon  become  injurious: 

!r..m  tt  effect  of  intense  light;    (a)  from  induced 

transpiration.  In  the  case  of  the  former,  superposition  will  give 
from  light-injury,  in  the  l.ittn  it  will  give  also  protection  from  d 
1 1  cnce  reduced  superposed  systems  become  characteristic  xerophvtk  vftfrts- 
tions,  and  the  ultimate  cases,  the  decussate  (a  +  a)  and  the  distichous  (i  -f  i), 
occur  as  very  general  limiting  phenomena  of  shoot-construction.  The 
intermediate  (i  +  a)  is  less  easy  to  check  when  the  i 
when  they  are  closely  packed  this  system  is  also 


by  '  torsion ',  as  in  Apicra  sftraJu,  Cyprus  alUrm/Mu.  and 

In  conclusion,  it  would  appear  that  Wiesner's  gmcraHtalioa  Is  to  bt 
taken  only  in  its  widest  significance  as  a  general  law  for  Und-v 
ng  far  back  in  the  phylogeny  of  the  race,  and  hence  so  deeply 
that  it  is  not  lightly  changed     Individual  apparent  exceptions  and 
cases  cannot  be  considered  merely  on  their  merits,  without  I 
thing  of  their  phylogcnetic  history.    Thus  whatever  be  the 

Ntcm.  the  existence  of  trimerous  and 
and    more  particularly  the   latter,   all   point   to  the 
Fibonacci   type  of  phyllotaxis  in   the  parent  stock  of 
apparently  as  the  solution  of  the  problem  of 


ancestral  forms  of  all  these  last  special  cases  must  have  been  origmalhr 
endowed  with  Fibonacci  phyllotaxis  in  the  vegetative  shoot,  however  BJMSCII 
it  may  have  been  modified  since.  The  general  problem  will  be  to  aceoos* 
for  this  phylogcnetic  feature.  It  is  also  evident  that,  in  approadiiag  the 
subject  of  the  mechanism  of  phyllotaxis-systcma,  the  Fibonacci  ratios  am 
of  primary  importance ;  all  other  types  of  construction  being  fSfSJdsd  as 
derivatives  of  this  parent  form. 

iay  be  objected  that  here,  as  in  other  purely  telcological 
.  ieMxrr's  theory  seeks  to  prove  too  much.  In  thus  e> 
r t  ucs  of  the  Fibonacci  ratios  under  maximum  illumination  by 
light,  it  has  little  to  say  on  the  relative  value  of  other 
closely  similar  at  first  sight  (anomalous  and  bijugate  systems), 
characteristic  of  plants  by  no  means  badly  illuminated  (/ftjMsau,  ( 
Sitf/tium).  Nor  can  a  whorled  (decussate)  construction  be  shown  to  bt 
objectionable  in  any  known  case  (Acfr.  Fr*xi**s.  Atscmhu)  of  shoots  Mil 
no  marked  xcrophytic  tendency.  All  compensatory  effects  avoid  any 
change  in  the  construction-system  at  the  actual  apex.  Nor  is  It  dear  I 
any  growing-point  could  become  conscious  of  the  tact  that  its 
were  ultimately  unsatisfactory,  or  know  how  to  aher 
be  the  correct  solution  of  the  problem !  C 


xerophytic  adaptations  are  put  in  where  the  iu|uWte 
mg  tissue-differentiation  in  Zone  III,  or  in 

•  R  J.M.L.  p.  15* 


22        On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 

Zone  II,  but  never  any  change  in  the  minute  growing-point  (Zone  I).  Nor, 
again,  is  vertical  light  such  an  essential  consideration ;  the  vast  majority 
of  plant-shoots  are  lateral  and  axillary,  obliquely  illuminated  at  their  first 
inception,  if  at  all ;  yet  no  distinction  can  be  drawn  between  the  apical 
construction  of  terminals  and  laterals.  In  a  few  examples  noted  (Silphium, 
DipsacHs)  the  terminals  arc  bi jugate  and  the  laterals  present  the  pu re- 
Fibonacci  organization.  Nor  can  any  such  considerations  apply  to  the 
entire  range  of  the  complex  subject  of  floral-phyllotaxis.  The  fact  that 
the  individual  never  'corrects'  the  phyllotaxis-construction  of  its  apex, 
whatever  the  external  illumination,  goes  to  show  that  teleological  inter- 
pretations applying  to  the  ontogeny  o?  the  individual  are  meaningless  ;  and 
though  it  may  be  more  plausible  to  regard  such  agencies  as  effective  in  the 
phytogeny  of  the  race,  there  is  little  evidence,  so  far  as  land-plants  are 
concerned,  of  any  mechanism  which  might  so  respond  to  conditions  of 
illumination,  any  more  than  it  does  at  the  present  time.  One  has  to  fall 
back  on  the  effect  of  natural  selection  acting  on  chance  mutations  in  the 
construction-system  ;  and  though  it  is  quite  possible  and  indeed  general 
for  plants  with  Fibonacci  phyllotaxis  to  deteriorate  such  construction  in 
every  possible  manner  (to  whorled,  anomalous,  bijugate,  and  wholly  irregular 
constructions),  there  is  nothing  whatever  to  show  how  it  may  be  possible 
for  the  happy  solution  of  the  Fibonacci  angle  and  ratio,  with  its  remarkable 
properties,  ever  to  arise  as  a  chance  mutation. 


IV 
OBJECTIONS  TO    THE   EQUIPOTENTIAL   THEORY 

ON  the  other  hand,  objections  to  the  equipotential  theory,  as  given  in 
precise  mathematical  form,  occur  naturally  to  the  botanist : — 

(i)  If  constructions,  as  postulated  in  terms  of  infinite  log  spiral  curves, 
are  equally  by  theory  impossible  in  the  case  of  a  plant  presenting  only 
finite  growth, — To  what  extent  can  the  angles  calculated  from  them,  or  from 
any  other  data,  be  regarded  as  quite  reliable ;  or  to  what  extent  may  they 
be  accepted  as  mere  approximations  ?  It  is  sufficiently  evident  that  the  rate 
of  active  growth  of  primordia  at  the  apex  of  a  plant-shoot  slows  down 
as  the  adult  stage  is  reached ;  and  such  retardation  is  the  necessary  accom- 
paniment of  plant-construction.  The  examination  of  such  retardation 
should  thus  have  an  important  bearing  on  the  general  phenomena  of 
phyllotaxis.  For  example : — As  the  entire  systems  are  considered  primarily 
from  the  standpoint  of  the  transverse  component  of  the  apex,  it  is  evident 
that  such  retardation  may  be  analysed  into  (i)  retardation  along  the  radial 
paths  of  the  section,  and  (2)  retardation  along  the  tangential  paths. 
Consideration  of  these  factors  separately  shows  at  once  that  no  radial 
retardation  can  affect  the  angular  divergences  between  members ;  since 
variation  in  growth  along  the  radial  paths  leaves  the  primordia  still 
travelling  outwards  along  the  same  radii  as  before,  but  at  slower  rates. 
On  the  other  hand,  any  retardation  along  the  tangential  circular  paths 
merely  pulls  the  surface  over  in  the  form  of  the  dome-shaped  apex  which  is 
the  familiar  accompaniment  of  shoot-development ;  and  as  the  transverse 
plane  is  alone  considered,  this  part  of  the  retardation-effect  vanishes  so 
far  as  present  purposes  are  concerned.  In  all  such  systems,  however  the 
growth  be  modified,  the  angular  divergence  remains  unaffected^  since  each 


Objections  to  the  Equifiottntiat  Tkany. 


>rdium  travels  radially  along  it*  ova  path.  This  is  the 
the  quuM-circlc  hypothesis ;  in  that  it  eliminate*  at  owe  all 
conceptions  of  •  Spiral  growth  Spiral  movement'.  .1  ,hc 

a  plant.    A  spiral  pattern  does  not  imply  spiral  movement ;  the 
of  growth  is  purely  radial ;  the  spiral  curves  of  the  pattern  being  mm*) 
retained  as  the  system  expands.    The  angles  calculated  for  log  spiral  data 

*.!_  .  .  i  i  •  i  f          .. 

swsssmsf  and 


are  thus  the  true 

the  tabulated  scries  at "once  shows 

to  the  Ideal  Angle.     So  dose  is 

little  doubt  as  to  the  suggestive  value  o?\Viesner's  Law.  and  that  spiral 

:isof  Fibonacci  ratios  has  been  acquired  phylogeoctka" 
aad  not  ontogenctically,  in  each  individual  case,  to  direct  response  to 


of  all  centrk  phytlotaxi.syslo»s,  and 
how  perfect  may  be  the  approximation 
the  approximation,  that  there  can  be 


problem  of  maximum  light-supply :  just  as,  on  the  other  hand,  tymoMtri 

tod  Law,  tends 


whorled  construction,  or  any  deviation  from  the 
approximate  the  solution  of  the  converse  problem  of 
tion,   which  is  quite  as  important  in  the  case  of 
exposed  to  intense  light. 

>  The  next  most  important  objection  is  the 
the  actual  means  whereby  such  mathematical  relations  ma'y  be 
the  growing-point  of  a  stem,  in  what  may  be  regarded  (omfet»i_ 
walls)  as  a  fairly  homogeneous  multinudeate  mass  of  protoplasm     HM 


problem  of  energy-distribution  in  such  an  apex,  only  suggested  by  the  theory. 
remains  -  iy  vague ;  yet  what  can  be  said  on  the  subject  may  be 

briefly  discussed  later.  At  the  present  stage  it  b  not  immaterial  to  point 
out  that  the  physical,  ionic,  and  molecular  relations  of  a  growing  mass 
of  protoplasm  probably  represent  the  most  infinitely  complex  construction 
in  nature,  and  any  facts  which  tend  to  throw  light  on  it  beyond  the  range  of 
visibility  should  be  welcome.  At  any  rate  the  problem  of  the  u«tua,in| 
of  a  phyllotaxis-construction  must  be  approached  sooner  or  laiet  the 
present  suggestion  merely  ceases  at  the  point  that  it  can  be  only  fdcmd  to 
manifestations  of  molecular  forces. 

>  Apart  from  the  actual  ontogeny  of  the  members  at  a 
point  in  the  rhythmic  sequence  denoted  by  a  phylh 
problems  arc  involved  in  the  initiation  of  the  systems 
(i)  at  the  apex  of  a  seedling. 
(a)  at  the  growing-point  of  a  lateral  branch. 
(3)  the  initiation  dt  novo  on  a  cell-surface,  as  on  the  leaves  of 

bulbifcrttm,  Nymfkat*  sp. 
This  section  of  the  subject  comprises  not  only  the  initiation  of  whoried 
and  spiral  constructions,  but  also  the  orif*t*ti*n  of  these  i;  stums  with 
regard  to  the  parent  axis.  Questions  of  the  initiation  and  orientation  of 
lateral  axes  have  been  very  largely  dominated  by  views  of  contact -orworc 
from  Schwcndcncr  (1875)  to  van  Iterson  (1907).  (cC  Webs*  in  Gocbefs 
Organography,  1900,  Eng.  Trans.,  p.  8a),  even  when  such  hypotheses  haw 
been  recognized  as  worthless  in  affording  any  explanation  of  the  comMO- 
tions  at  the  main  apices:  this  being  largely  due  to  the  fret  thai i  in  so  many 
familiar  cases  of  axillary  bud-development  the 
ch  confined  quarters  that  the 


to  arise   in  such  confined  quarters  that  the  influence  of  the 
adjacent  appendages   is  sufficiently  suggestive  to  the  eye. 
many  difficulties  of  such  a  view  may  be  mentioned  :— 
(i)  The  actual  initiation  of  new  centres,  which  mar  " 
growth-effect  can  be  MM  as  '  Icmf-primordia ',  does  not 
dearspace.    Aiillary  bods  which  devttop  to  posfcfcs*  in 
may  be  wholly  wanting,  as  on  rapttjeatsadmf  shoots  of 
present  constructions  which  diner  in  no  respect  from   ' 
compacted  buds. 


24        On  the  Interpretation  of  Plienomena  of  P/tyllotaxis. 

(2)  Identical  contact-relations  are  not  followed  by  similar  resulst.     Thus, 
in  lateral  branches  of  Armn  Literal  buds  arise  in  the  axils  of  certain 
leaves  of  a  system  (8+  13),  Riving  two  lateral  orthostichies  of  lateral  branches  in 
a  frondose  form.    The  contact-conditions  are  identical  for  every  bud  on  either 
side  of  the  main  shoot,  yet  the  bud-constructions  are  R.  and  L.  image-pat 
without  rule.     (Fig.  XI). 

(3)  Theories  of  contact-pressure  ignore  the  remarkable  phyllotaxis-con- 
structions  characteristic  of  the  ontogeny  of  flower-buds.    The  fact  that  the  vast 
majority  of  floral  shoots  are  orientated  with  the  fourth  member,  or  sepal  2  of 
the  quincuncial  calyx,  'median   posterior,'    shows  that  some  positions  may 
become  constants,  with  possibly  subsequent  biological  benefit.     Yet,  (a)  the 
Papilionaceae  retain  sepal  i  (the  third   member)  median  anterior  with  great 
reliability ;   (b)  the  twin-image  patterns  (R.  and  L.)  are  very  approximately 
equally   distributed,   apart   from    the    phenomena  of   dichasial    construction ; 

1c)  while  the   majority  of  flowers  have   sepal    2   'median   posterior',  others 
cf.  Lobelia)  present   the   inverted   twin-image  (sepal  2,  anterior)  with   equal 
constancy,  under  apparently  identical  conditions  of  origin. 

(iv)  The  next  most  important  question  concerns  the  conceivable 
phylogeny  of  the  mechanism.  This  difficulty,  again,  must  be  approached 
sooner  or  later:  the  present  paper  concludes  with  a  few  remarks  on  the 
more  obvious  features  of  the  problem  ;  more  particularly  as  affecting  the 
relation  of  the  equipotential  theory  to  '  Land  Flora ',  and  the  general 
features  of  other  systems  of  space-form  and  ramification  which  do  not 
come  under  the  same  ideas  of  mechanism. 

(v)  Lastly  may  be  mentioned  the  most  serious  objection,  common  to 
all  mathematical  considerations  of  phyllotaxis-problems,  and  the  one  which 
appeals  most  directly  to  every  botanist,  not  so  much  to  those  with  a  distaste 
for  exact  figures,  as  to  those  possessing  a  healthy  scepticism,  in  dealing 
with  plant -protoplasm,  as  to  whether  it  is  ever  possible  for  a  plant  to  work 
in  any  precise  mathematical  manner  which  can  be  readily  formulated. 
This  has  been  admirably  expressed  by  Sachs  as  the  error  of 'gratuitously 
introduced  mathematics'.  In  fact,  why  introduce  mathematics  at  all? 
To  what  extent  do  any  mathematical  relations  whatever  exist  in  the  plant  ? 
and  to  what  extent  is  the  introduction  of  mathematical  calculations  or 
theory  to  be  justified,  either  from  the  standpoint  of  descriptive  morphology, 
the  actual  physiological  mechanism  of  production,  or  the  utility  of  the 
schemes  in  biology. 

The  classical  generalizations  associated  with  the  names  of  Schimper 
and  Braun,  divested  of  all  extraneous  theoretical  deductions,  reduce  to  the 
very  elementary  fact  that,  on  looking  round  the  plant-kingdom,  as  a  matter 
of  simple  observation,  the  great  majority  of  plant-shoots  show  a  remarkable 
tendency  to  the  repetition  of  certain  numbers,  as  seen  on  counting  the 
patterns  in  which  the  leaf-systems  are  expressed.  The  most  frequent  of 
these  numbers,  put  down  on  paper  in  numerical  order,  may  be  written  i,  2, 
3.  5»  H,  13,  21,  and  the  series  may  be  extended  to  34,  55,  89,  144,  in  less 
frequent  examples,  but  no  further.  This  is  a  fact  of  botanical  observation  ; 
and  when  to  this  is  added  the  fact  that  the  contact-parastichies  are  always 
indicated  by  a  successive  pair  of  these  numbers,  it  is  legitimate  to  deduce 
the  conclusion  that  it  is  the  peculiar  ratio  (approximately  constant)  indicated 
by  these  Fibonacci  numbers,  which  is  the  essential  point  of  the  whole  story. 
This  is  all  that  the  numbers  give,  and  all  that  is  required  for  Wicsner's  Law. 
The  superposed  edifice  of  '  Ideal  angles', '  Other  series',  '  Complementary 
series ',  '  Divergence  angles ',  '  Fractional  expressions ', '  Orthostichies ',  &c., 
are  so  much  mathematical  elaboration,  more  or  less  purely  irrelevant, 
which  adds  nothing  whatever  to  the  simple  facts  contributed  by  the  plant ; 
and,  as  we  have  seen,  such  mathematical  'playing  with  figures',  as  Sachs 


Objections  to  the  EqmpolmtM  Tktory.  a$ 

termed  it,  soon  wen  iy  off  the  track  with  the  ^T^ltirn  of  the 

helical  constructions,  originated  by  Calaiuirtni  (1754),  whkh  cannot  deal* 
OM:  the  facts  of  ontogeny  at  a  growing-apex.  In  the  MOC  way, 
even  Wiesner's  hypothesis  was  originally  ^^-frfd  as  a  mathe 
proposition  solely  on  these  mathematically  deduced  fraction*  The 
law  follows  readily  from  the  properties  of  the  Fibonacci 
method  of  approaching  it  through  the  Hl^ 


can  have  no  bearing  on  the  plant,  rendered  it  obscure; 
\Vie.M  ,»  was  pleased  to  show  by  this  means  that  the"'  Main 

of  divergence-fractions  would  be  more  advantageous  than  any  other 
for  purposes  of  minimum  overlapping.  The  'Other  series'  are  purely 
imaginary  conceptions,  built  up  along  absurd  idealistic  lines  of  thought, 
many  of  which  are  still  to  be  text-books.1  Similarly  the  log  spTrai 

notation,  although  going  back  to  umerals,  is  at  first  sight  rcaoaWd 

so  complicated  by  the  addition  of  lit  tie- known  spiral  constructions*  that 
it  only  adds  further  difficulty  to  a  bewildering  mass  of  figures.  The  new 
'divergence  angles'  rather  add  to  the  contusion  than  otherwise.  For 
example,  did  any  botanist  in  his  senses  ever  believe  that  the  scales  of 
a  Pine-cone,  with  a  phyllotaxis  expressed  in  the  books  a*  'A*,  arc 
actually,  or  were  ever  really  laid  down  at  a  divergence-angle  of  A  of 
360°  =  137°  8'  34-28"  from  one  another;  the  cone  being  obviously  not 
a  cylinder,  ami  the  orthostichies  not  straight ;  or  is  it  made  much  dearer 
by  saying  that  the  (84-13)  conc  nas  *Ls  scales  really  at  a  dUrn^im)  of 
137°  30'  38".'  Yet  if  they  were  not,  what  is  the  good  of  introducing  the 
theory  ?  Or,  again,  do  we  believe  that  the  disk-florets  of  a  '  |jL '  Suonower- 
head  were  laid  down  at  a  divergence  angle  of  137°  31'  41-1  -".  while  those 
of  a  'j^fV  head  were  accurate  to  137° 30'?*  Can  we  even  form  a  mental 
picture  of  the  accuracy  of  such  angles,  or  do  we  stop  to  think  what  is  meant 
by  one  minute  of  arc  in  the  case  of  a  small  circle  ?  *  On  the  other  hand,  as 
already  indicated,  if  137°  30'  28-936"  is  the  ideal  angle  tor  exposing  a  given 
number  of  leaves  for  purposes  of  photosynthesis,  accurate  to  seconds  and 
a  number  of  decimal  places,  why  are  so  many  plants  satisfied  with  .. 

of  144°  exactly.     Again,  taking  log  spiral  notation,  is  it  to  be 
supposed  that  the  leaves  of  a  (2  +  3)  system  are  instead  acosn 

s°  27'  42",*  as  already  suggested  ?    Are  these  things  accurately' 
or  if  not,  to  what  extent  may  they  be  approximately  true,  or  be 
as  true  \\ithin  reasonable  range,  and  what  is  reasonable  range? 
advance  in  phyllotaxis-thcory  can  be  made,  these  questions  require  to  be 
answered  and  placed  on  a  sound  footing.     If  our  conceptions  of  phyOoti 
are  now  given  as  based  on  orthogonally  intersecting  log  spirals  and  it 
been  admitted  that— 

log  spirals  never  occur  in  any  pla: 

(2)  the  orthogonal  construction  can  never  be  moisnrad,  and  is  not 

to  obtain ; 

(3)  if  the   postulated  divergence-angles  are  not   true  either,  < 

be  measured ;  the  whole  subject  appears  so  hopelessly  futik 
wonders  why  it  was  ever  put  forward,  or  what  any  one  could 
seen  in 
However,  the  object  of  the  present  paper  is  to  show  that  the  Equ 


cells 

me  r  isle  m 
rhrn.it  ii  .il 

R.P.M.L.,  p.  340. 


26        On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 

Theory  has  sufficient  evidence  in  its  favour  to  survive  all  these  destruc- 
tive criticisms,  since  it  is  itself  a  constructive  hypothc  i  Its  value  con- 
sists in  affording  a  first  step  in  a  subject  which  so  rapidly  increase 
complexity  that  without  such  a  guiding  conception  the  problem  could 
never  be  disentangled.  To  take  a  simple  illustration : — An  ordinary  tree- 
trunk  with  cambial  increase,  adding  a  ring  of  wood  every  year,  can  be 
considered  as  working  out  a  series  of  concentric  circles,  and  the  original 
pith  should  remain  the  centre  of  the  circumference  indicated  by  the  old 
trunk.  This  is  a  simple  geometrical  conception ;  but  no  botanist  supposes 
that  any  old  tree-stem  cut  across  would  ever  yield  a  mathematically  exact 
circle,  or  that  the  pith  would  ever  be  in  the  geometrical  centre.  One  might 
be  pleased  to  find  a  specimen  approximating  a  system  of  concentric  circles 
to  illustrate  an  elementary  lecture,  but  that  is  all.  On  the  other  hand, 
what  happens  is  this: — the  simple  geometrical  conception  remains  as 
a  standard  of  reference,  useful  enough  in  its  way,  and  following  from  simple 
premisses.  It  is  clear  that  if  these  premisses  alone  were  concerned,  the  plant 
would  work  on  the  postulated  geometrical  form  ;  but  the  problem  becomes 
increasingly  complicated  by  secondary  factors:  the  value  of  the  standard 
of  reference,  however,  remains  unaffected.  The  Equipotential  Theory  has 
been  put  forward  primarily  as  such  a  Standard  of  Reference,  in  dealing 
with  phyllotaxis-constructions  as  seen  in  transverse  sections  of  the  growing- 
points  of  plant-shoots.  The  mathematical  consequences  of  such  systems 
have  been  deduced  and  tabulated  definitely  from  a  mathematical  standpoint : 
evidences  have  been  obtained  from  plant-constructions  which  suggested  that 
such  a  course  was  justifiable  and  worth  doing.  After  all,  we  do  have 
Fibonacci  numbers  (among  others) ;  we  do  have  spiral  patterns,  and 
a  growing  system ;  these  phenomena  require  to  be  explained.  However, 
as  previously  indicated,  the  point  at  which  the  ordinary  botanist  hesitates 
is  the  marvellous  precision  of  the  so-called  '  divergence  angles '  to  several 
places  of  decimals  of  a  second,  which  are  all  deduced  mathematically,  and 
not  measured  on  the  plant  at  all.  This  may  give  an  aspect  of  pseudo- 
scientific  accuracy  to  the  text-book  presentation  of  the  subject,  but  are 
these  things  really  so  ?  Is  the  plant  accurate  or  not ;  to  what  extent  are 
we  justified  in  demanding  scientific  accuracy,  which  may  be  non-existent. 

The  standpoint  taken  up  here  is  sufficiently  straightforward.  One  is 
not  justified  in  assuming  any  accuracy  in  the  plant  whatever,  beyond  what 
can  be  measured^  or  fairly  assumed,  as  in  the  case  illustrated  by  the  circular 
tree-trunk.  Deductions  for  practical  purposes  (e.g.  to  illustrate  Wiesner's 
Law)  must  be  taken  from  actual  plant-specimens.  A  living  plant  is 
preferable  to  Wiesner's  models  of  photographic  paper  set  at  angles  only 
roughly  approximated  by  hand.  Calculations  involving  minutes  and 
seconds  are  useless  when  it  is  difficult  to  guarantee  accuracy  to  several 
degrees.  However  much  mathematical  conclusions  are  useful  and  suggestive 
as  a  standard  of  reference  to  keep  by  one,  there  is  no  justification  for  going 
beyond  data  actually  obtained  from  plants  themselves.  Preferably  the 
matter  may  be  made  clearer  by  keeping  the  mathematical  data  strictly 
on  one  side,  and  the  plant-data  on  the  other.  What  exactly  does  the  plant 
give,  and  what  questions  are  to  be  looked  up  on  the  mathematical  side? 
To  do  this  a  method  is  required ;  and  again  it  is  in  establishing  such 
a  method  that  the  utility  of  the  Equipotential  Theory  becomes  at  once 
apparent,  as  affording  the  essential  standard  of  reference,  which  serves  not 
only  as  guide  to  the  mode  of  operation,  but  suggests  the  working-methods 
to  be  employed.  Nor  is  this  merely  'seeing  in  the  plant-apex  what  we 
want  to  see'.  The  proof  of  the  utility  of  a  theory  is  found  in  the  fact 
that  it  serves  to  bring  out  points  which  would  otherwise  have  escaped 
observation. 


GENERAL  METHOD  FOR  THE  EXAMINATION  OF 
PHYLLOTAXIS-PHENOMENA 

IP  all  mathematical  data  are  merely  mathematical 
useful  as  a  Standard  of  Reference,  but  never  directly 
plant ;  if  orthostichy  lines  are  merely  properties  of  systems  of  spirals  of 
Archimedes,  which  only  obtain  with  members  of  equal  volume  or  equal 
radial  depths ;  if  intersecting  contact -parastichies  can  be  only  traced  m 
a  few  standard  examples,  as  for  instance  the  Pine-cone,  in  which  practically 
no  secondary  compensatory  disturbances  have  affected  the  original  pattern.— 
it  remains  to  consider  to  what  extent  it  may  be  possible  to  obtain  any  really 
reliable  method  for  dealing  with  actual  plant-construction*.  It  It  so  far 
obvious  that  the  growing-point  at  which  the  system  Is  '  laid  down '. 
becomes  visible,  is  the  only  region  concerned.  Whatever  secondary 
come  into  operation,  and  however  invisible  the  first  causes,  it  is  not 


to  go  behind  the  first  appearance  of  the  primordia  themselves  on  the  actual 
arched  or  flat  growing-point  surface ;  while  this  region  can  be  only  dealt 


with,  to  any  degree  of  accuracy,  by  carefully  cut  sections.  It 
therefore,  to  obtain  a  fairly  satisfactory  method  for  the  eiramlnallne  of  an 
actual  apical  system.  For  this  purpose  a  fairly  large  bud-apex,  with  veil- 
marked  simple  Icaf-primordia  and  little  longitudinal  extension  in  the  main 


axis,  b  to  be  preferred.  Sections  of  hardened  spirit- material,  cut  by 
are  subsequently  treated  with  potash  and  Eau  de  Javelle  to 
distortion  and  bring  the  object,  as  near  as  possible,  to  the  normal 
A  drawing  of  such  a  section,  grazing  the  smooth  apex  of  the 
point  in  Semf*ri>ivum  calcaratum  is  given  in  Fig.  XII.  On  an  ordinary 
pencil  drawing,  the  original  figure  being  7-8  inches  In  dismetrr  (Zcsss,  A., 
Oc  3,  Cam.  Lucid),  angles  can  be  measured  to  half  a  degree,  and  linear 
dimensions  to  half  a  millimetre.  Of  the  various  errors  likely  to  be 
duced,  the  most  important  are:  (i)  shrinkage-effects  In  the 
(2)  the  difficulty  of  keeping  the  section  exactly  transverse ;  (5) 
of  estimating  the  exact  centre  of  the  system. 

The  figure  illustrates  a  broad  circular  apex 
which  pass  over  into  areas  with  rhomboklal  section :  so 


-7-a.    There  is  nothing  particularly  remarkable 
about  the  manner  in  which  it  is  obtained: 


several  cases  the  leaf-primordia  are  wider/  spaced  (1-4  »»-X 
parallel  corves.  Such  extensive  shrinkage  vffl  (sttodoc*  so^ 
error,  when  calculations  are  made  to  han  a  mfflinwiis,  that  fart 


measurement  is  needless.    There  is  no 


gro\\ 


28        On  the  Interpretation  of  Phenomena  of  Fhyllotaxis. 

No  further  evidence  of  the  production  of  a  phyllotaxis-system  is  available : 
the  method  adopted  is  simple  but  imperfect :  the  point  is  to  see  if  it  is  sufficient 
for  the  purpose.    It  being  quite  clear  that  what  is  wanted  is  a  complete  under- 
standing of  the  whole  of  the  facts  given  by  one  apex  ;  and  ihis  can  be  done 
better  on  a  large  drawing  than  by  observing  the  image  in  the  microscope. 
The  fir>t  thing  to  do  is  to  measure  up  the  figure,  and  collect  all  the 
data— the   length  and  breadth  of  the  members,  the  angle  subtended  by 
each,  and  the  angular  divergences  between  successive  numbers.     The  last 
is  rendered  possible   in   the  case  of  older  primordia,  in  which  the  first 
production  of  protoxylem  gives  a  central  point  to  the  member,  with  con- 
siderable accuracy.     The  youngest   leaves  (o,  i,  2,  3,  4)    can    give   only 
approximate  values. 

The  subjoined  table  includes  the  results  checked  for  this  particular 
apex  (the  figure  being  a  reduced  copy  of  the  original,  and  not  necessarily 
accurate),  and  the  following  features  may  be  noted : — 


i 

II 

III 

IV 

V 

VI 

UJ 

%. 

- 

« 

h  8  5 

*i 

Number 
leaf. 

ill 

o^S 

H 

1 

J3  c  6 

*ll 

n\ 

0 

_ 

I  : 

* 

*»••* 

i 

12  mm. 

15  mm. 

1-35 

39° 

137-5°  ) 

136°    \ 

2 

14 

'7 

1-21 

38 

U5 

w    «    V 

137-5 

^  °"i 

3 

4 

15 

35 

\'& 

59 
56 

137-5 
142 

<  l&E 

134 

143-8 

^% 

I 

17 
19 

30 

35 

1-76 
1.84 

61 

67 

127        t 
M3 

135-8' 
136-7 

129     } 
140-3 

136-06° 
136-93 

I 

9 

10 

28' 

39 
46 

53 

55 

2-00 

1-9' 
1-82 
2-29 

66 
76 

i! 

137-8 
133-7 
'45 

1  37*36 
136-3 
136-9 
137-7 

133' 

M3 
133 

137-33 
136-92 
136-76 
137-56 

11 

32 

73 

3-35 

92 

140 

137*3 

138-8 

137-26 

12 

31 

75 

2-42 

86 

137-4 

139-4 

.  24 

»3 

35 

86 

2-46 

90 

132 

137-26 

|£J 

'24 

14 

44 

100 

3-37 

96 

143 

136-86 

141'3 

136-9 

15 

39 

102 

2-62 

90 

136 

137-86 

137-3 

i    7-76 

It 

45 

1  JO 

2.66 

95 

136-9 

137-2 

\l 

46 

135 

2-72 

90 

137*3 

136-8 

136-68 

Apex  of  Se mperui vum  calcaratum  (3  +  5),  Fi^.  XII. 

(i)  The  pattern  is  a  well-marked  (3  +  5)  construction,  since  9, 14, 17, 1 2, 
or  4,  7,  9,  12,  for  example,  give  a  rhomb  of  contact :  the  longer  *  3 '  curves 
are  clean-edged,  while  the  '  5 '  curves  are  very  slightly '  stepped '  after  about 
two  cycles.  On  looking  up  the  table  for  such  a  system,1  the  angle  sub- 
tended by  a  quasi-circlc  primordium  should  be  61°  44'.  That  subtended 
by  a  full  quasi-square  rhomb,  85°.  The  divergence  angle  is  137°  38'  5o".2 
The  ratio  of  the  length  to  the  breadth  of  the  quasi-circle  is  i :  i  -004. 

At  first  sight  the  standard  of  reference  does  not  appear  to  give  any  very 
great  assistance. 

(a)  Circular  tangential  paths  drawn  through  the  central  bundles  of  the 
older  leaves  indicate  that  these  leaf-primordia  are  obliquely  orientated  :  this 
being  associated  with  the  assumption  of  a  more  rhomboidal  form  as  they 


Rel.  Phyll.  Mech.  Laws,  pp.  338,  340. 


8  loc.  cit.,  p.  340. 


Method  for  Examination  of  Phyh 


are  pressed  into  close  contact  relation*     That  it  to  My.  • 
primordia  tend  by  contatt-prtssurt*  $9  become  *  y>«rf  sysw* 

(3)  The  youngest  primordU  are  rounded,  but  distinctly  I 
extended  than  the  theoretical  quasi-drcles,  Na  i  giving  a  ratio  o< 
instead  of  the  theoretical  1-004 :  «•    That  U  to  say.  the  youogtm  vfefelc' 
primordia  are  already  possibly  radially  reduced  in  growth;  and   men 
reduction  extends  progressively  until  at  No.  17  the  ratio  of  breadth  to 
radial  depth  is  a-;a :  i.     This  implies  that  thtrt  u  *  /nyrwijis*  r+ 
retardation  in  the  rat*  of  growth  throughout  thi  entire  rytiem.  ml  i/ 
already  set  in  when  the  pnmordia  first  appear  at 


As  the  angle  subtended  at  the  centre  (Column  IV)  also 
increases,  it  would  seem  that  there  is  an  increase  in  the  rate  of  U 
extension ;   but   this  is  probably  largely  a  subjective  effect ;   since  ihc 
primordia  are  gradually  pressed  into  the  form  of  cc 
angle  subtended  by  which  would  be  85°.    Other 
operation  in  dealing  with  the  tangential  dimensions  of 
may  be  cut  at  a  slightly  higher  level  than  their  successors :  the 
beyond  85°,  as  a  matter  of  fact,  is  not  very  considerable  in 

(5)  In  the  first  column  the  succession  of  radial  dime 

increase  in  exact  progression ;  and  the  same  applies  to  CoL  II  for  the 
tang<  \tension :    there  are  again  inequalities  in  the  ratio  of  these 

measurements  as  expressed  in  Col.  III.  Such  inequalities  are  dearly  the 
expression  of  the  error  of  the  section,  especially  from  the  standposst  of 
obliquity  in  the  exact  transverse  plane.  This  is  again  *—pi»MfTfff  in 
Col.  IV.  The  angles  subtended  by  Nos.  3  and  1 1 .  for  example,  are  i 
too  large  ;  so  to  a  less  extent  are  those  of  14  and  16.  Assuming  that 
young  leaf- primordia  are  broader  below,  it  would  appear  that  the 
dips  a  little  on  the  top  left-hand  corner. 

(6)  Since  the  angle  normally  subtended  by  a  primordium  filling  fcs 
place  in  such  a  contact-system  should  be  61-44°.  it  would  appear  that  the 
members  indicated  as  i,  a,  3,  &c.,  are  smaller  than  they  should  be  if  the 
mechanism  of  the  pattern  were  determined  solely  by  conditions  of  balk* 
ratio.     This  again  eliminates  the  standpoint  that  the  men  n:e  *f  the  p*\me* 

:  as  it  first  appears  has  any  necessary  connexion  with  the  temttrmtiem  to 
be  worked  out.     In  fact,  the  angles  subtended  by  the  youngc 
would  even  suggest  the  initial  points  of  a  (5  +  8)  system,  the 
which  normally  subtend  30°   10'.     On  the  other  hand,  the  sj 

taken  as  it  stands ;  and  it  is  called  u  because  the  n 

the  members  are  in  this  con  tact- relations).  *  soflKdeatly  obrow  «••* 

number  of  patterns  seen  varies  according  to  rates  of  growth.  In  thit 
particular  case,  for  example,  the  older  leaves  of  the  bud  may  proa*  a 
definite  (;,  .ingcment,  auite  distinct  from  *' 

(7)  Special  interest  attaches  to  the 


protoxylem-points  for  13  leaves,  and  approximated  from  1-4-    Connp 

errors  arc  at  once  indicated,  rendering  any  very  exact  use  of  the  n 

very  doubtful :  for  example,  Col.  V  gives  a  range  between  1*7*  and  MjT; 

Col.  VI  (with  a  difference  of  i  mm.  on  the  drawi 

of  the  system)  a  range  of  ia9'-i43-*°-    Thc  lruc 

the  latter  than  the  former.    As  5,  H,  10,  I J  are 

14  conspicuously  high,  the  position  of  these 

they  are  affected  by  the  inequality  of  the 

errors  of  centring  and  the  obliquity  of  the 

taking  averages. 

1   IOC.  CSL,  Cf.  JX  244- 


3O        On  tke  Interpretation  of  Phenomena  of  Phyllotaxis. 

Thus  the  average  of  the  entire  set  of  1 7  leaves  (Col.  V,  VI)  gave  137-11° 
and  137-04°  respectively,  as  opposed  to  a  theoretical  angle  of  137-65°. 

But  since  any  successive  5  members  in  such  a  system  should  make 
mathematically  similar  contact-cycles  around  the  stem,  it  is  interesting  to 
take  averages  of  members  successively  5  at  a  time.  On  doing  this  it  will 
be  found  that  these  averages  range  from  135-8°  to  137*86°  for  the  first 
column,  and  between  136-06°  and  137-76°  for  the  second.  Or,  omitting  the 
fir>t  members,  for  which  the  central  points  were  only  approximated,  the 
average  of  a  cycle  of  5  is  always  remarkably  well  within  one  degree  of 
the  ideal  angle. 

There  is  thus  sufficient  evidence  in  this  apex  of  a  remarkable  approxima- 
tion to  a  divergence-angle  of  about  137°,  in  a  system  which  is  at  the  same  time 
undergoing  radial  retardation,  and  subject  to  mutual  contact-pressures  which 
tend  to  produce  a  rhomboidal  contour  in  the  members. 

(8)  Even  more  remarkable  is  the  fact  that  the  divergence-angle  remains 
equally  constant  throughout  the  scries,  so   far  as  can  be  measured.     As 
opposed   to   Schwendenerian   theories  of  displacement,  there   is  not   the 
slightest  indication  of  any  '  lateral  displacement '.     No  member  has  slipped 
out  of  its  relative  position  owing  to  the  effect  of  hypothetical  contact- 
pressures.     The  error  of  the  Dachstuhl-hypothesis  is  at  once  demonstrated 
by  systematic  measurement. 

(9)  It   is   also  evident  that   in   an   Archimedean   system,  as   that  of 
Schimper  and  Braun,  which  would  be  attained  when  the  adult  members 
show  equal  radial  depth,  the  divergence- angle  must  become  3/8  of  360° 
=  135°.     But  so  far  in  this  system,  which  covers  still  growing  members, 
there  is  no  sign  that  such  a  secondary  divergence-angle  is  being  produced. 
While  again  there  is  sufficient  evidence  that  the  members  here  are  not  yet 
attaining  the  equal  depth  of  the  adult  condition  (Column  I),  and  that  such 
mathematical  relations  do  not  obtain. 

(10)  The  fact  remains   that,  allowing  for  the  errors  of  the   section, 
a  system   of  measurements   may   be   sufficiently  accurate   to  reasonably 
demonstrate  that  the  angle  of  137°  is  very  fairly  and  uniformly  approxi- 
mated by  the  plant ;  and  that  the  angle  is  practically  a  constant,  however 
the  system  may  vary  its  growth-phenomena  in  other  respects. 

The  same  method  may  be  applied  to  other  plant-apices  with  closely 
similar  results.  One  other  example  may  be  taken,  of  an  apex  which 
differs  in  many  respects  from  that  of  Scmpervivum.  Tips  of  Cobaea  scandens 
are  of  interest  as  affording  examples  of  leaf-systems  in  which  practically  no 
contacts  obtain  between  the  young  members  ;  and  yet  each  young  leaf 
stands  well  away  from  its  neighbours,  with  its  own  well-marked  angular 
divergence.  Section  shows  that  these  primordia  are  rapidly  differentiated 
and  segmented  into  compound  laminae,  but  still  remain  well-spaced  apart. 
There  are  no  contact-pressures,  no  squeezing  into  quasi-square  rhombs,  and 
hence  no  slipping  of  the  angles ;  while  the  exact  bilateral  symmetry  of  each 
leaf  with  regard  to  a  radius  passing  through  the  central  point  of  the  vascular 
system  is  a  most  striking  feature  of  the  construction.  (Fig.  XIV.) 

A  similar  camera  lucida  drawing  shows  leaves  in  a  (3  +  5)  system  numbered 
1-14,  and  the  divergence-angles  are  measured  as  before.  The  youngest  pro- 
tuberances are  still  indefinite,  and  leaves  with  the  protoxylem  indicated  range 
from  7  onwards.  Omitting  14,  which  has  been  already  displaced  in  making  the 
preparation,  as  shown  by  its  asymmetry  with  regard  to  the  radial  plane,  the  average 
of  cycles,  taken  5  at  a  time,  is  137-36°,  137-8°,  and  137-2°  respectively.  The 
general  accuracy  of  the  method  is  thus  in  close  general  agreement  with  that 
found  in  the  case  of  Sempcrvivum,  though  over  a  more  restricted  field,  and  again 
indicates  an  undoubted  approximation  to  the  Fibonacci  angle  of  137^°,  or  to  the 
theoretical  angle  of  the  construction-system  (34-5)  =  137-65°. 


Method  for  Examination  of  Phyltotaxit-pkcnamem* 
C*l**  ttmJmt.  Plf .  XIV. 


NO.  offer. 


i-a 


4-5 


8-9 
9-io 

10-11 

ii-i2 


.3 


ft 


I3S» 

134 

«33 

«33 


•37* 
'37-i 


There  is  no  need  at  this  stage  to  multiply  further  fffiaplfi  It  fe 
sufficient  to  state  that  similar  results  are  the  general  property  of  plant- 
apices  with  constructions  expressed  by  such  low  numerals  a*  (2+  t)  and 

•5). 
But  it  may  now  be  taken  as  demonstrated  that : — 

(1)  There  is  a  very  definite  approximation  in  suck  nstewu  99  m  *nrb 
of  I37°>  or  I37i°;  which  for  convent***  may  be  labelled  the  T^rTfJUl 
Angle*  of  flant-tonstnutiotis.     While  the  Ideal  Angle  of  Werner  b  137* 
30'  28-936  ',  and  the  angle  given  by  the  Equipotential  Theory  far  a  (J  +  3) 
construction  is  137°  38*  50". 

(2)  On  the  other  hand,  there  is  no  evidence  mrilaUt  m  the  pUmt  ft 
present  of  any  closer  approximation  to  these  data*  which  remain  those  of 


a  standard  of  reference.    The  plant  gives  an  angle  «flffn'ffiHly  ftTTWitf  to 
suggest  that  both  these  latter  conceptions  may  be  correct ;  but  there  b  no 
need  either  to  postulate  or  to  imagine  absolute  agreement  or  precision  in 
instruction.     It  may  be  so,  but  there  is  no  mean 

nstrating  the  fact.    The  data  given  by  the  plant  and  tb< 
data  of  the  calculated  standards  require  to  be  kept  perfect 

No  greater  accuracy  of  measurement  can  be  obtained  in  a 
construction,  which   is  regarded  as  presenting  exact  circular  Sf 
The  smallest  consideration  shows  that  the  circles  of  a  plant-apex  arc 
mathematically  accurate,  and  the  spacing  of  the  member*  of  the  whocit  b 
only  approximated,  however  beautiful  the  effect  may  be  to  the 
Here  again  mathematical  calculations  of  the 
a  standard  reference.     Once  mathematical  < 
into  such  a  subject,  a  curious  obsession,  that  since  the 


are  so  exact,  the  plant  substance  b  necessarily  obliged  to  work  in  an 
precise  manner,  becomes  a  general  (act  of  belief  ex 


_  try  difioalt  ID 

eradicate-. 

(3)  Still,  though  there  b  no  evidence  that  more  accurate  antics  are 
attained  in  the  plant-construction,  this  mural  ajfvmrimatfm  a  me  aqgie 
10  w  tk<  grtat  outstanding  f«t  *f  F*~™  pkyOrtms*  Thb  angle  ft* 
obtained  from  the  plant  by  observation  only:  other  phmomrm  awocuted 
with  it  are  subsidiary,  and  confirm  the  idea  that  the  constancy  of  the  as*le. 
within  quite  a  small  range,  b  the  central  feature  of  the  entire  Mbjea,  and  it 
the  fact  which  has  to  be  accounted  for. 


On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 

The  problem  remains,  therefore^ — How  is  this  angle  adjusted  or  obt 
at  the  plant-apex  f     Where  does  it  come  from  phylogenetically  f  and  above 
all — By  what  mechanism  can  it  be  maintained  in  a  growing  snoot-system  f 


VI 
THE   MECHANISM  OF   PHYLLOTAXIS 

AN  angle  of  approximately  137$°  has  been  termed  the  Fibonacci 
angle,  in  contradistinction  to  the  '  Ideal  Angle '  of  the  Schimper-Braun 
notation ;  the  latter  a  purely  mathematical  abstraction,  while  the  former 
is  an  established/^/  of  observation  taken  directly  from  plant-constructions. 
The  value  of  this  angle  is  so  peculiar,  that  no  reasonable  person  can  further 
refuse  to  believe  that  it  actually  represents  an  approximation  in  the  plant- 
organization  to  the  theoretical  Ideal  Angle  (137°  30'  28-936")  which  would 
afford  maximum  illumination  to  the  leafy  system  if  vertically  displayed  ; 
and  that  this  is  no  mere  coincidence,  but  a  phenomenon  of  such  wide 
occurrence  that  it  must  undoubtedly  afford  some  clue  to  the  remarkable 
problems  of  shoot-construction.  But  such  phenomena,  as  expressed  in  the 
constancy  of  the  angle,  even  if  no  more  accurate  than  the  angle  accepted 
(of  about  1374°),  require  a  mechanism  for  their  production  ;  and  it  is 
naturally  in  this  mechanism  that  the  whole  of  the  physiological  interest 
of  the  subject  is  centred.  One  gets  little  further,  for  example,  by  supposing 
that  an  originally  irregular  development  of  aimless  'enations'  settles  down 
to  a  rhythmic  process;1  because,  in  spite  of  the  *  Strobilus-Theory ',  there 
is  no  sign  of  such  a  proceeding  in  land-vegetation.  An  indefinite  pro- 
tuberan  ce  (provided  for  botanically  as  an  '  emergence ')  has  not  necessarily 
the  very  striking  and  peculiar  attributes  associated  with  a  *  leaf-appendage  ;a 
an  adventitious  shoot  is  an  organization  of  a  quite  different  category; 
while  it  may  be  noted  that  the  mechanism  for  producing  one  *  enation ', 
as  also  even  one  pseudopodium,  must  be  quite  distinct  from  the  mechanism 
which  involves  a  serial  repetition  of  the  act.  It  is  in  fact  this  rhythm 
which  demands  the  essential  mechanism  ;  and  the  evolution  of,  or  the 
necessity  for,  such  rhythm  is  the  fundamental  feature  of  the  problem. 
The  question  remains,  therefore,  as  to  where  such  a  mechanism  can  be 
seen  or  traced. 

In  regard  to  this,  at  the  outset,  one  point  may  be  granted  as  firmly 
established  : — the  mechanism  has  no  relation  whatever  to  the  more  obvious 
cell-framework  of  the  plant-apex.  It  is  only  necessary  to  examine  a 
longitudinal  section  of  such  a  growing-point  to  see  that  there  is  nothing 
visible  beyond  the  dividing  cell-meristems.  New  primordia  arise,  as  seen 
in  section,  as  waves  of  lowest  elevation,  often  involving  a  considerable 
number  of  cells  from  the  first;  there  is  no  sharp  demarcation  of  such 
undulations,  nor  can  it  be  said  where  to  a  single  cell  each  exactly  begins 
or  ends.  The  protoplasm  of  the  apex  may  be  preferably  regarded 
as  a  mass  of  fluid  colloidal  plasma,  in  which  the  secondary  production 
of  denser  colloidal  cellulose  films  may  have  but  little  effect  on  the  physical 


1  Bower  (1908),  '  Origin  of  Land-Flora,'  p.  141. 

*  R.P.M.L.,  p.  190:  a  remarkable  case  being  presented  by  the  form  and 
arrangement  in  alternating  whorls  of  the  emergences  on  a  syncarpous  ovary  among 
Palms  of  the  section  Lepidocaryinae  :  Rahia  Ruffia  (6  +  6). 


Tki  M«ka*i*m  of  Pkyllotmxu. 


33 


condition  of  the  living  and  fluid  mass  as  a  whole.    The  i 

of  some  Vascular  Cryptogams,  in  which  the  cdUiniu  nay  be 


in  one  rhythmic  and  spiral  sequence,  apparently  rtnmiaated  by  a  afcsjie 

•       •  •  a          »   •  •          •         •        ^  »  /  » 


apical  cell,  while  the  leaf-members  arise  either  in  a  wholly 

sequence  (Asputiam),  or  even  in  symmetrical  circles  (whorls  of 

in  which  the  apical  cell  still  independently  cuts  off  ••••ill  to 

sequence  of  three  rows,  again  apparently  controlled  by  the  oriental 

tin  rotating  nuclear  spindle  of  the  apical  cell  ifsrif    sho«s  at  once  that  the 


mechanism   underlying   leaf-production   in  higher  Land-Flora  miM   be 
something  quite  distinct  from    cytological 


The  mechanisei  of 


Fibonacci  phyllotaxis  being  thus: — 

(i)  Sufficiently  accurate   to   attract  attention  as  being   restricted  to  a 

divergence-angle  of  about  137°. 
(a)  Completely  independent  of  cell-segmentation. 
(3)  Due  to  some  wholly  invisible  cause  beyond  the 

obeervatioa 

Some  suggestion  has  to  be  made  as  to  what  it  may  be;  evea 
more  be  gained  than  a  working-hypothesis. 

If  the  mechanism  b  then  invisible,  there  is  really  no  eacaf 
conclusion  that  it  must  be  in  some  way  molecular,  in  the 
crystallization,  for  example,  may  be  termed  molecular.  This  ia  the  next 
natural  step  to  consider :  when  molecular  properties  (ail,  it  may  be  time 
to  pass  on  to  something  even  more  abstruse.  But  at  this  point  the  botanist 
who  is  interested  in  things  seen,  or  which  can  be  treated  experimentally, 
has  to  stop.  Molecular  mechanisms  still  wait  on  the  physicist.  The 
botanist  has  to  be  content  with  a  working-hypothesis  which  will  indede 
all  the  facts  of  observation.  The  first  point  at  issue,  therefore,  is  how  to 
account  for  this  constant  approximation  to  a  divergence-Angle  of  af 
137°.  Once  the  standpoint  of  cell-control  is  eliminated,  the  jjusAll 
of  explanation  open  to  the  botanist  are  obviously  icsUkted  The 
moat  natural  view  to  discuss  is  that  of  the  traditional  *  Genetic  Spiral'. 
Is  it  conceivable  that  a  plant-apex  of  undiffercntutcd  meristDBl  has  the 
power  of  measuring  off  angles  of  about  137}°  at  Mated  intervals  with 
a  considerable  degree  of  accuracy  and  constancy,  and  that  the  inlirqisaf 
pattern  is  the  result  of  two  such  associated  factors— one  of  exact  angular 
measurement,  and  the  other  involving  a  time  sequence  ?  An  angle  of  I  JT§* 
is  not  easy  to  obtain  by  human  geometrical  methods,  and  there  is  no  etsji 
how  it  may  be  measured  in  the  plant.  It  b  of  course  possiblr  that  sack 
might  occur;  but  if  this  be  the  case  the  problem  remains  fc°P*j*»  < 
further  solution,  since  there  is  no  trace  of  any  mechanism  wnkh  wfll 
produce  the  given  effect.  It  the  ideological  11  nnraHiafinas  of  Wiesnar 
held  to  such  an  extent  that  the  Fibonacci  angle  was  Lua^esiory  far  el 
shoots  devoted  to  photosynthesis,  it  might  be 
angle,  having  been  attained  at  some  c" 
natural  selection,  to  remain  constant 
tion,  and  so  heritable.  But  this  is  not 
all  but  one  of  many.  Shoots  of  the  same  plants 
accurate  Fibonacci  systems  will  give  other  pattenis  ^based^  OP 
constant  divergence-angles ;'  the  angle  may  be  ra  " 
constant  until  changed  again.  That  is  to  say, 
hypothetical  simple  mechanism  that  would  always  give  the  seal 


distant  period,  had  become  feed  by 
t  or  ingrained  in  the  pbafrniaeriie 
ot  the  case:  the  angle  tj7ft*  b  a*V 
e  same  plant*  which  otherwise  beg 
ve  other  patterns  based  on  egsjesy 

lf  ma v  Kr  ififtflflv  ckeSMBsV  VSJt  naTBBfll 


of  1 374°  in  all  plants  ;  but  we  cannot  so  readily  imagine  a  steph 
that  would  equally  well  repeat  almost  any  angle  with  eqml 


1  R.P.M.Upp.  101,  104; 
vi  c 


34       On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 

In  fact  a  merely  cursory  examination  of  the  general  problem  of  phyllotaxis 
among  subaerial  vegetation  is  sufficient  to  show  that  the  divergence-angle 
can  have  no  primary  causal  relation,  but  is  now  itself  the  consequence  of 
other  factors,  following  as  a  mathematical  property  of  the  numerals  involved 
in  the  spiral  pattern,  as  expressed  by  the  number  of  parastichics,1  whatever 
may  have  been  its  original  significance. 

Again  the  data  presented  by  the  apex  of  Sempcrvivum*  for  example, 
about  which  there  can  be  no  doubt  whatever,  arc : — the  system  is  a  growing 
and  expanding  one,  producing  leaf-members  (i)  at  a  fairly  constant 
divergence-angle  of  137}°,  and  (2)  presenting  a  set  of  curves  making 
a  pattern  in  which  3  paths  cross  5,  i.  e.  (3  +  5).  If  the  former  phenomena 
be  dismissed  as  too  vague  for  practical  consideration,  one  can  still  fall  back- 
on  the  latter.  There  is  no  further  choice.  Is  such  an  angle  of  137-5° 
obtainable  from  any  possible  construction  involving  the  numbers  3  and  5  ? 
Yes,  it  is  a  problem  of  uniform  mathematical  growth,  the  solution  of  which 
states  that  orthogonally  intersecting  logarithmic  spirals  equally  spaced  in 
this  complementary  ratio  will  give  successive  intersections  at  137°  38'5o";3 
and  this  is  the  simplest  solution  possible.  We  are  therefore  bound  to  take 
it  as  the  next  standpoint  for  consideration :  if  it  fails,  further  discussion 
will  be  necessary. 

The  Equipotential  Theory  simply  accepts  this  mathematical  generaliza- 
tion as  the  basis  for  further  elaboration ;  and  suggests  that  nothing  more 
is  required.  The  problem  is  solved  once  for  all.  Instead  of  a  wholly 
miraculous  mechanism  repeatedly  measuring  off  angles  of  137^°,  the 
proposition  reduces  to  a  question  of  2  relatively  simple  numerals  involved 
in  a  special  but  visible  geometrical  construction. 

Even  so  the  question  of  mechanism  remains  sufficiently  wonderful :  but 
a  simple  analogy  may  render  the  suggestion  a  little  clearer.  Few  features 
are  more  remarkable,  yet  always  accepted  without  question,  as  a  common- 
place of  elementary  botany,  than  the  initiation  of  the  protoxylcm  points  in 
a  typical  Dicotyledonous  root.  Nor  is  the  mystery  underlying  this  pro- 
blem rendered  any  clearer  by  suggesting  that  the  few  protoxylems  of  such 
a  root  are  the  specialized  relics  of  a  polyarch  xylem,  or  of  a  uniform 
exarch  xylem-cylinder.  Centres  are  isolated  in  small  and  variable  num- 
ber at  approximately  equal  distances,  in  a  special  region  of  the  stele,  and 
these  control  not  only  the  subsequent  anatomical  differentiation  of  the 
root,  but  its  system  of  subsequent  ramification.  Analysis  of  this  com 
paratively  simple  phenomenon  shows  that  4  factors  are  involved,  which 
must  be  actually  represented  in  the  apical  meristem  :— 

(1)  A  power  of  numerical  choice.     The   number   is   fairly  constant, 
though  within  certain  well-defined  limits ;   one  numeral  being  selected  by 
each  apex :  e.  g.  2,  3,  4,  5,  or  6  in  many  common  plants  (Pinus,  Quercus). 
We  do  not  know  how  the  apex  has  the  power  to  count ;    or  how  the 
number  is  regulated  ;   but  the  numeral  for  each  root  is  a  fact  of  simple 
observation. 

(2)  A  factor  for  equal  spacing.     Whatever  numeral  may  be  selected, 
the  individual  points  are  approximately  equally  spaced  around  a  circular 
path :  i.e.  2  at  180°,  3  at  120  ,  4  at  90°,  &c.    We  do  not  know  to  what  extent 
these  angles  are  accurate  at   the  moment  of  initiation,  or   only  roughly 
approximated.     As  there  can  be  little  advantage  in  accurate  spacing,  it 
is  usually  considered  ideologically  that  the  arrangement  is  only  approxi- 
mate.    But  both   the  arrangement   and  the  spacing  are  again   facts  of 
observation. 

1  R.P.M.L.,  pp.  218,  234.  *  loc.  cit.,  Sempervivum,  pp.  16,  244. 

»  loc.  cit.,  p.  340. 


The  Mechanitm  of  Ph\lbUadt. 


(3)  The  fatten  expressed  by  such  a  series  of 
spaced  round  the  periphery  of  the  stele,  m  really,  tern 
standpoint,  an  orthogonal  system  to  which  radii  intersect  a 

right  angles  '.he  impulses  of  xylem-dHfcreatietioo  proceed 

petally  at  right  angles  to  the  surf  ace  of  the  stele. 

(4)  This  remarhable  delimitation  of  new  o 
restricted  to  a  definite  region  of  the  root,  and  so 

IKKsible  to  define  the  pericycle  to  exact  histological 
to  th 

Now  comparison  of  the  phenomena  to  be  explained  at  the 
a  stem  in  the  initiation  of  a  phyl 


is  sufficiently  close  to  be  very  striking.    The  same  4  fectors  are  involved 
in  both  cases :  the  only  difference  being  that  the  case  of  the  shoot -apex  Is 


somewhat  more  elaborate  geometrically,  but  not  to  any  other 

Thus: — 


The  power  of  numerical  choice  involves  a  usually  quite  simple 
(a)  Equal  spacing  b  provided  by  the  geometry  of  log-spiral  sysi 
may  be  continued  with  the  growth  of  the  plant  •  to  infinity '. 

(3)  The   resultant  pattern  involves  orthogonally  ' 
curves,  of  which  the  radius  and  drdc  are  only  I 

(4)  The  production  of  new  centres  b  restricted  to  surface-layers  of  the 
(epidermis  and  cxo-cortcx).  less  defined  than  to  the  case  of  the  root ; 
while  these  centres,  once  initiated,  are  subsequently  utilised  to 

the  space-form  of  secondary  ramifications. 

It  is  difficult  to  avoid  the  general  conclusion  that  the  growing 
has  a  power  of  numerical  choice,  within  certain  limits,  and 
constant ;  as  also  a  faculty  for  equal-spacing  to  order  to  give  a 
result.     How  b  it  done?    We  can  so  far  form  no  real  opii 
mechanism  b  beyond  the  range  of  visibility,  and  hence  we  sisBfl 
.is  a   fact  of  observation  dent  that  a  fixed  o 

4  impulses ',  of  some  description,  must  radiate  from  the 
certain  equally-spaced  lines  of  distribution,  which  affect  a  region  of  < 
tissue-units  only.     That  b  to  say,  the  path  of  the  impulse  is 
spaced*  and  limited. 

Very  closely  identical  phenomena 
the  shoot-apex  -.—impulses  of  definite 
to  narrow  paths,  apparently  radiate  from  the  growth-tent* 

ctions,  and  unth  exactly  equally  balanced  distribution*  as  in 
case  seU  /  one  direction  and  <  in  the  other.     The  points  ml  warn* 

these  impulses  intersect  give  the  initial  points  far  the  c*mmtmt*ment  a/ 

fral  appendage.     The  impulses  are  invisible,  but  the  efttt  at  At  /mmt 
of  intersection  is  seen  in  the  initiation  of  a  new 
restricted  to  the  surface-layers  of  the  apex.     This  new  ce> 
wholly  from  the  parent 
an  entirely 
umnmmism 
(factors  of 
selection  which  has  given  the  leaf-number 

Geometrical  considerations,  comparable  With  those  which 


u  must  be  similarly  charac 
number,  t^nalh-spaced  and 
?om  tne  growta^entu ,  tn  At* 


from  the  par***  grtwth-ctntre  in  it*  «***«** /<r*ri.  It  <mm+ta 
irely  new  set  of  growth  phenomena ;  some  At  **fe*nim  a/  sir 
ism  of  the  correlation  of  the  mm  frimtnton*  with  its  parent  em** 
s  o  the  omasMrcte).  others  the  inherit  p~*mct  of  the  j**t  mitm* 


say  that  an  equal  spacing  of  protoxy lem  to  a  tetrarch  roc 
90°  apart,  again  enable  us  to  deduce  the  geutneUkal  c 
follow  from  a  mathematically  perfect  system  of  equal 


as  an  error  of  5-10°  in  the  case  of  a  root  protoxykm 
excite  any  special  remark,  so  there  b  no  reason  to  postulate :  « 
mathematical  spacing  of  these  impulse  paths,  since  they  cannot  be 

ca 


36       On  th£  Interpretation  of  Phenomena  of  rhyllota\ 

by  observation ;  the  use  of  the  mathematical  discussion  is  again  solely 
that  of  affording  a  standard  of  reference.  The  exact  application  of  the 
Theory  of  Equipotential  should  now  be  clear.  With  regard  to  the  power 
of  choice,  as  expressed  by  a  numerals  in  the  shoot-apex,  there  is  nothing 
to  be  said.  It  is  even  more  vague  than  that  of  the  single  choice  made 
by  a  root-apex  ;  in  that  it  may  be  at  any  time  changed  by  the  addition 
or  loss  of  paths  giving  the  great  variation  possible  in  changing  phyllotaxis- 
systcms.  Yet  we  know  by  observation  the  possible  and  probable  range 
of  such  choice,  and  that  it  is  commonly  restricted  to  the  lowest  ratios.1 
Nor  do  we  know  what  is  meant  by  '  impulses ' ;  whether  classed  as  '  paths 
of  distribution ',  '  lines  of  force ',  *  paths  of  nervous  tension ',  or  any  other 
vague  expression  with  which  ignorance  may  be  clothed.  But  the  mechanism 
of  equal  spacing  can  be  studied  mathematically,  and  it  is  given  in  perfect 
form  by  the  equipotcntial  construction  in  terms  of  orthogonally  intersecting 
spirals.  It  is  thus  open  to  argument  that  it  is  from  a  certain  demand  for 
equal  spacing  that  the  orthogonal  construction  follows;  and  that  the 
recognition  of  a  geometrical  construction  comparable  with  spiral  vortex - 
construction,  or  with  propositions  of  electric  potential,  is  here  paralleled  by 
the  plant  from  a  purely  geometrical  standpoint,  rather  than  as  indicative 
of  any  fundamental  agreement  in  an  expression  of  distribution  of  either 
kinetic  or  static  energy.  This  in  fact  is  the  first  objection  of  the  physicist. 

Propositions  of  equal  spacing  can.  however,  be  as  readily  worked  out  for 
any  angle  of  intersection  so  long  as  the  angle  remains  constant.2 

The  critical  point  lies  in  the  fact  that  the  angle  in  actual  growing 
plant-constructions  is  never  maintained  as  a  constant,  and  in  the  possibility 
that  a  system  of  orthogonal  construction  may  be  significant  of  some  deeper 
fundamental  property  of  protoplasm.  The  whole  point  of  the  preceding 
investigation  has  been  centred  in  an  attempt  to  prove  the  significance  of 
this  property,  by  assuming  its  existence  and  deducing  the  consequences/5 
Added  to  this,  the  remarkable  agreement  of  the  fundamental  recognized 
properties  of  a  leaf  with  the  mathematical  properties  of  the  quasi-circle, 
even  when  inscribed  in  the  orthogonal  mesh  of  an  asymmetrical  con- 
struction, has  been  put  forward  as  a  sufficient  proof  of  the  truth  of  the  idea; 
since  no  other  simple  mathematical  generalization  can  give  such  a  result. 
On  the  other  hand,  it  has  been  suggested  that  orthogonal  construction 
merely  implies  the  utilization  of  the  simplest  angle  to  keep  adjusted,  and 
that  the  geometry  of  the  construction  is  perhaps  the  end  of  the  mathematical 
possibilities.  But  even  here,  such  a  view  cannot  represent  the  whole  truth  ; 
since  in  dealing  with  a  living  mass  of  protoplasm  we  still  have  to  account 
for  the  possibility  of  the  evolution  of  such  a  mechanism  in  the  plant  From 
such  a  phylogenetic  standpoint  it  is  scarcely  possible  that  such  a  geometrical 
system  could  have  been  originated,  unless  it  had  in  some  way  utilized 
features  of  construction  which  were  pre-existent,  as  some  attribute  of  the 
molecular  forces  of  growing  protoplasm. 

In  the  narrowest  sense,  then,  the  equipotential  theory  remains  as 
a  standard  of  reference  for  the  rules  of  equal  spacing :  in  a  wider  sense, 
still  to  be  hoped  for,  as  put  forward  when  the  proposition  was  first 

1  R.P.M.L.,  cf.  p.  342. 

'  Misled  by  an  enthusiasm  for  drawing  pretty  but  meaningless  figures,  Van 
Iterson  has  in  this  way  attempted  to  improve  on  the  log-spiral  constructions 
originally  put  forward  as  sufficiently  satisfactory;  regardless  of  the  simple  fact 
of  observation  that  in  no  growing  system  in  a  plant-shoot  do  the  parastichy  curves 
continue  to  intersect  at  any  constant  angle !  G.  van  Iterson,  jun.,  '  Blattstellungen,' 
1907. 

'  R.P.M.L.,  p.  230. 


Th*  Mcfkaxism  of  Pkyllot** 


enunciated,  it  may  give  a  clue  to  the 

growth,  and  prove  in  fact  the  fir*  efficient  Hep  in  the  progress  o*  4* 


effective  plant -morphology.    While,  however,  these  IT  mum 
remain  very   much  in  the  air,  it   is  so  far  important  to  not* 
mathematical  and  geometrical  considerations  are  to  bt  regarded  soldy  as 
a  standard  of  reference  for  phenomena  which  may  be  exact  rnougii  far 
t)u  biological  purposes  of  the  plant,  but  are  not  necessarily  absolute. 
that  is  no  reason  why  geometrical  constructions  should  not  be 

i  onlcr  to  widen  the  scope  of  the  inqi 

Above  all,  it  is  essential  to  attempt  the  linssl^slliiii  of  the 
'.ogenetic  origin  of  such  a  remarkable  serial  of  phenomena.    Intn 

growth-factors  of  subacrial  vegetation  founded  on  hypothesis 
of  subaerial  evolution  from  Cryptogams  of  fresh-water  ponds  have 
singularly  unfortunate  in  other  departments  of  Botany.    The 
remains  that  the  phyllotaxis-mcchanism  of  the 
plant  of  the  land  is  after  all  but  the  highly 
of  some  much  older  if  more  obscure  ancestral 
present  conditions  of  environment  it  was  never  designed  to 
it  is  understood  that  biologically  every  factor  of 

.ibly  an  adaptation,  possibly  the  end-term  of  some  prrrolinj, 
of  successful  adaptation  in  the  long-continued  nrogicssion  of 
mere  ideological  interpretations  of  su 

the  attempt  should  be  made,  if  possible,  to  trace  such  phenomena  of 
organization  to  first 


VII 

rilYLLOTAXIS-PHENOMENA    IN    CRYPTOGAMS 
AND   THALLOPHYTA 

RHYTHMIC  expressions  of  leaf-arrangement  are  by  no  means  confined 
to  higher  plants  or  Phanerogams,  the  Seed-Plants  of  more  dominant  land 
Flora.    Very  elegant  applications  of  the  general  principles  are  to  be  sun 
among  Pteridophyta,  in  which  leaf-members  are  similarly  arranged  on  ar~ 
of  a  sporophytc-generation  ;  and  it  is  sufficiently  Jlll±jlA11* tfc-t  •*••  **•* 


the  soil-habit  by  the  first  land-plants,    On  the  other 
involving  rhythmic  spiral  succession  are  equally  cha 

Bryophyta  of  the  Moat-series  (Bryineae);  though  in  this 

mena  arc  to  be  traced  in  the  gamctophytfc  (sexual)  generation,  in 
owing  to  the  inefficiency  of  the  absorptive  lire  nanism,  the  full  an* 


-  to  the  inefficiency  of  the  absorptive 
is  of  a  transpiring  leaf-lamina  are  never  attained ;  y*  it  is 
old  from  such  seriate  appendages  of  the  Moss-axis  the  mot 
c  of  leaf-members  in  these  vestigial  relics  of  more  archaic  tn 
types.    The  case  is  again  complicated  \  act  that  very  ****** 

spiral  arrangements,  in  which  Fibonacci  symmetry  majr  be^an 
obtain  in  the  case  of  many  of  the  more  massive  Br»r~ 
phyceae-Fucoideae),  in  the  orientation  of  the  more  or 
like  lateral  ramuli ;  leaving  little  doubt  that  the  phyil 
in  fact,  a  still  older  function  of  the  axis  of  marine  types  of 


38        On  tJie  Interpretation  of  Phenomena  of  Phyllota,\ 

that  the  presentation  of  such  phenomena,  even  in  a  more  elaborated  and 
special  form,  can  be  but  the  continuation  and  amplification  of  factors  of 
marine  phytobcnthon ;  and  that  it  is  to  the  sea  that  one  must  look  for  the 
origin  and  primary  intention  of  this  remarkable  relation.  Kvcn  behind  the 
horizon  of  the  more  massive  parcnchymatous  marine  alga,  it  is  possible  to 
trace,  even  back  in  the  plankton-phase,  distinct  expressions  of  spiral  organ- 
ization associated  with  Fibonacci  ratios  ;  not,  it  is  true,  among  autotrophic 
phytoplankton,  but  all  the  more  remarkably  in  that  similar  appearances 
may  be  exhibited  in  holozoic  and  animal  forms  of  life  among  the  great 
group  of  the  Foraminifera. 

I.  Pteridophyta.  The  tendency  of  the  general  run  of  Filicineae  to 
produce  their  leaves  in  spiral  sequence,  as  also  predominantly  in  terms  of 
ratios  of  the  Fibonacci  series,  may  be  accepted  without  discussion  as  a 
simple  fact  of  observation.  The  example  of  the  apex  of  Aspidittm  /•'///>- 
Mas?  has  been  adduced  as  affording  the  most  perfect  expression  of  the 
complete  independence  of  the  individual  primordia  of  such  a  sequence  from 
contact-pressures  in  early  stages,  as  also  for  their  derivation  from  the  tissues 
of  the  apex  without  any  regard  whatever  for  the  3-sided  apical  cell,  or  the 
spiral  of  its  successive  segments.  Variations  on  the  theme  may,  however, 
supervene,  as  :— 

(1 )  Irregularities  in  construction,  as  expressed  by  the  addition  (or  often 
loss)  ol  construction- curves  in  the  system,  exactly  in  the  manner  of  irregu- 
larities in  the  axes  of  Cactaceae  and  Cycads,  and  similarly  the  expression  of  a 
deteriorated  mechanism.     In  large  tree-ferns  the  curves  may  be  *  anything ' ; 
and  in  special  cases  it  is  possible  to  trace,  by  the  equalization  of  the  curves 
in  either  direction,  the  attainment  of  symmetry,  general  or  partial  only, 
giving  a  precise  whorled  effect  (Alsophila)* 

(2)  In  cases  of  advanced  dorsi ventral  organization,  the  construction 
may  reduce  to  (i  + 1),  with  the  plane  of  bilaterality  correlated  with  dorsi- 
ventral  symmetry  to  give  two  rows,  right  and  left  of  the  axis,  familiar  in 
the  type  of  Pteris^  clearly  the  highest  expression  of  advanced  specialization, 
and  by  no  means  '  primitive*. 

(3)  In  the  highly  specialized  dorsiventral  shoots  of  Hydropterideae, 
whorled  symmetry  may  be  associated  with  the  dorsiventral  habit,  involving 
a  special  condition  of  orientation  and  leaf-presentation,  as  in  the  essentially 
decussate  (2-1-2)  system  of  Azolla,  and  the  remarkable  (3  +  3)  construction 
of  Salvinia  with  marked  heterophylly  in  the  whorls.8 

In  spite  of  advancing  specialization,  or  general  decadence,  the  type  of 
the  Filicineae  may  be  said  to  be  primarily  based  on  Fibonacci  symmetry. 
More  marked  is  the  case  of  Equisetum,  in  which  a  similar  3-sided  apical 
cell -mechanism  is  associated  with  whorled  symmetry  in  leaf-production, 
giving  3  members  to  a  whorl  in  the  limiting  case,  but  readily  rising  to  over 
30.  Variations  in  the  case  of  E.  Telmatcia^  the  finest  available  form,  have 
been  recorded  (loc  cit.,  p.  147),  while  the  occasional  production  of  spirally 
constructed  shoot -systems,  over  the  whole  or  a  portion  of  their  length,  as  a 
4  monstrosity ',  has  been  shown  to  follow  the  mechanism  of  the  simplest 
variant  from  symmetry,  as  the  case  of  the  (m  :  m  +  i)  ratio.  It  may  be 
accepted  that  the  symmetrical  constructions  of  E  guise  turn  are  wholly 
secondary,  and  are  associated  with  extreme  xerophytic  habit,  giving  a 
sequence  of  alternating  whorls  when  the  construction  remains  constant  for 


1  '  Relation  of  Phyllotaxis  to  Mechanical  Laws  ',  p.  29,  Fig.  35. 
5  Ornamental  vases  being  cut  from  a  stem  thus  presenting  a  symmetrical  pattern 
in  the  sclerosed  plates  associated  with  the  stelar  skeleton. 
3  R.P.M.L.,  loc.  cit.,  p.  280. 


Pkyll*t*dt~pk*»mm*  in  Crypugmmt  and  Tk*U*pkjU.     39 


(•finite  number  of  interoodes.    That  this  special  case  of  v 
mechanism  is  of  extreme  antiquity  follows  from  the  constancy  of  'the 
whorled   construction   in  all  fossil  records  of  allied   Equtsetteeac.    The 
secondary  regression  to  a  spiral  freak  is  exactly  paralleled 
sperms  in  the  cases  of  ntppuns  and  C^sjautrtiM. 

more  elementary  types  of  shoot-construction,  cC  LjapMam  md  /j 

>gh  decussate  symmetry  also  widely  obtains,  and  may  be 
with  an  advanced  phase  of  secondary  dorsiventrality,  as  in  the 
of  .SY/arijfs/Af,  as  V^p^d  on  the  earlier  spiral  constniction  of 
Illustrations  of  the  extreme  range  of  numerical  schemes  in  the  case  of  the 
deca  opodium  Sclago  have  been  given  (R.P.M.L..  loc.  dt,  F%* 

issociation  of  (5:5).  - ',  and  (a :  a),  rmphadrinf  the 

readiness  with  which  such  symmetrical  variations  follow  from 
as (5 :  6),  (4 :  5).  of  the  (m  :  m+  i) order:  the  i 


<>f  Fibonacci  ratios,  as  seen  in  the  case  of  other  xerophytk  types,  cf 
carpus  (loc  cit.,  Fig.  104)  and  Scd*m  (loc.  cit,  Fig.  104). 

Of  special  interest  again  are  examples  of  spiral  constniction  cut  i« 
calcitc  sections  of  calcified  coal-balls  of  the  Carboniferous  Epoch. 
really  accurate  transverse  sections  of  the  actual  apex  may  be  rate,  aad 
ill  be  decomposed,  sufficient  evidence  is  forthcoming  to  show 
Fibonacci  relations  prevailed  at  this  early  period,  quite  as  well  as  to-4 
and  the  construction  may  be  as  readily  interpreted. 

Thus  Slopes  ('Ancient  Plants,'  1910,  p.  136)  figures  i 
with  curve-construction  very  dearly  expressed  of  the  4:7:11 
apex  is  naturally  somewhat  damaged,  but  the  type 
secondary  '  anomalous '  ratio. 

Scott  ('Studies  in  Fossil  Botany,'  i,  1908,  p.  137)  gives  a 
young  stems  of  Ispidodt*dron  Har«mrhi,  with  the  vascular  to 
the  outer  cortex  retaining  their  spiral  orientation  in  a 


may  be  expressed  as  (9:14).  again  only  a  minor  variation  oa  (813) 
again  clearly  referable  to  such  deteriorated  mechanism  as  may  bs  fouad  SB  a  I 
fern  or  Cycad-axis. 

An  interesting  example  from  an  unnamed  LtJ&ttritmt  axis  (cat  by 
Bot  MUL.  Oxford,  No.  43)  illustrates  very  clearly  the  iuns  SJSMSS.  as  espwjssd 
by  the  remains  of  ihe  trace-bundles  of  the  macerated  susm,  of  ihs  fans  (8 
the  Fibonacci-ratio  being  definitely  present  as  an  iiniawlilihli  esawfftr  of 
occurrence ;  and  the  phyllotaxis  of  such  a  cone  may  be  so  dssufcsd  ss  teem  of 
further  evidence  of  the  apical  consliuction  (Fig.  XV). 

On  the  other  hand,  it  is  interesting  to  note  thai  the  dsstffeutk*  of  i 
bundles  of  an  axis,  although  affording  a  curv*paiawa  to  icrm*  of  UM 
employed,  does  not 


ployed,  does  not  necessarily  give  the  actual  uaisuaaioa  *i  tW  growty 

rtte^ 
a.    Thus  a  slide  of  L+***,*  ktodT   kat  b   Plrot  Oim.  UaK 


p.  325).    Thus  a  slide  of  L&Mtm+m  (kttxfly  tea*  by  Plrot  < 
Collection,  I^*ttr4m*  fihau)  shows  quite  distinctly  the  trac^b 
axis,  in  spite  of  partial  maceration,  in  a  system  which  would  bt 
(7:  n);  but  the  section  nearest  the  actual giuaiaf  petal  show 
primordia  very  distinctly  in  a  coatactcerks  of  (3  :  4>.  ihousj 
the  periphery  of  the  section  to  (4:7),  and  sillsiiliry  (7 :  \i\ 
rate  of  growth  in  the  leaf- members. 

Probably  the  actual  apex  of  all  UyMudswJioua  wat  lafinaiij  to 

lively  low  numbers,  just  ss  in  the  case  of  modem  Gj  •*•*•»«••  (el. 

.),  and  there  ii  no  need  to  wpkr  uno  «»V>^**X+ 

divcrgence^tctioiis,  deduced  fronAhe  uunridimina  of  ths  haflaai 

older  axes.    (Scott,  loc.  dL,  p,  118.) 

Thus  allowing  a  margin  fee  (i)  secondary  asMsaptMM  < 


4O        On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 


(2)  secondary  attainment  of  dorsivcntrality,  (3)  deterioration  of  the  mechanism, 
either  in  non-photosynthetic  strobilus-construction,  or  in  deteriorated  and 
reduced  axes, — all  phenomena  largely  associated  with  extreme  biological  adapta- 
tions for  special  and  usually  xerophytic  environment,  or  the  elaboration  of  re- 
productive shoots  of  limited  growth, — the  Pteridophyta  appear  to  be  as  con- 
stant to  the  general  mechanism  of  leaf-production  and  the  ratios  of  Fibonacci 
orientation  as  are  the  corresponding  ecological  and  floristic  developments  of  higher 
Angiospcrm  vegetation ;  and  the  phenomena  of  primary  Fibonacci  orientation 
are  to  be  traced  unchanged  at  least  as  far  back  as  the  beginnings  of  Vascular 
Plants. 

II.  Bryophyta.  The  case  of  the  Bryophyta  runs  on  distinctly  different 
lines,  in  that  the  leaves  are  produced  by  the  apical -cell  mechanism  of  a 
gametophyte-axis,  now  dominant  and  controlling  the  segmentation  of  the 
apex  both  into  cells  and  members.  The  mechanism  is  still  essentially  that 
of  the  same  3-sided  apical  cell  as  found  in  Filicineae  and  KquiVtincac,  and 

nilarly  the  expression  of  the  rotation  of  the  nuclear  spindle  in  succc 
mitoses,  as  affording  the  minimum  condition  of  three-dimensional  (centric) 
organization.  Such  a  mechanism,  if  perfect,  would  be  expected  to  give 
leaf-members  in  3  vertical  orthostichies,  arranged  with  a  divergence  of  120°. 
This  may  be  fairly  approximated  in  a  few  examples,  as  in  the  case  of  the 
submerged  moss  Fontinalis,  growing  in  weak  light  with  an  elongated  axis, 
with  the  leaves  on  the  adult  shoots  running  in  3  only  slightly  curved  '  spires '. 
In  the  more  normal  type  of  the  Eubryales  the  leafy  Moss-shoot  presents 
characteristic  Fibonacci  symmetry,  and  formulae  for  leaf-arrangement  have 
been  commonly  tabulated  in  the  fractions  of  the  Schimper-Braun  notation, 
and  even  in  anomalous  scries,  though  it  must  be  said  on  very  slender 
evidence.  Thus  Muller  (1894)1  gives  Fontinalis  as  '  \  divergence ',  and  the 
adult  effects  of  others  produced  by  *  torsion ',  as  commonly  *  § '  (Sphagnum), 
|  (Funaria,  Bryum>  Mnium),  ^j,  ^,  ^f,  as  also  fractions  of  other  series 
as  $,  ^-,  but  never  whorls.  Polytrichum  commune  is  scheduled  -jSj,  P. 
pilulifernm  2\,  and  P.  formosum  Jf . 

The  effect  in  the  adult  axis  may  be  more  readily  understood  by  cutting 
a  transverse  section  of  the  stem  of  P.  commune,  and  noting  the  orientation 
of  the  trace-bundles  of  the  leaves,  in  the  manner  described  for  Lepidostrobus. 
The  figure  (Fig.  XVI)  illustrates  the  fact  that  the  trace-bundles  pass  verti- 
cally down  the  axis  in  3  curve-systems  corresponding  to  the  three  lines  of 
successive  segments  ;  but  an  apparent  *  compensation '  enables  them  to  be 
spaced  out,  and  the  divergence-angles  may  be  very  approximately  measured 
by  taking  the  radial  median  line  of  the  leaf-trace  area.  In  this  case  about  a 
dozen  members  may  be  fairly  accounted  for,  affording  10  successive  divergence- 
angles  of  135,  140, 131,  i43<  134,  i3°*5> 145-.5, 129.  >5*5»  ™7'5  ;  the  average 
being  136-8°,  with  an  individual  range  to  129°  and  1.52*5° ;  or  taking  the  outer- 
most better-expressed  areas  three  at  a  time  to  avoid  errors  of  centring,  135-3° 
and  135°  :  that  is  to  say,  a  very  suggestive  Schimper-Braun  angle,  as  ex- 
pressed by  the  fraction  £  of  360°,  but  equally  to  be  regarded  as  the  approxi- 
mate derivative  of  a  Fibonacci  construction, in  which  a  system  in  Archimedean 
notation  is  attained  secondarily  with  the  equalizing  of  the  internodes  of  the 
adult  axis.  The  mechanism  of  'compensation'  is  introduced  apparently 
in  the  actual  apical  cell  itself,  or  its  immediate  derivatives,  to  the  extent 
that  the  segment-walls  are  not  parallel  with  that  of  the  parent  cell,2  and  this 
feature  suggests  directly  that  the  Fibonacci  condition  of  symmetry  is  the 
primary  factor  after  all,  and  the  mechanism  of  a  3-sided  apical  cell  is  adapted 

1  In  Englerand  Prantl  (1894),  p.  177. 

*  Mtiller,  loc.  cit.,  p.  178:  Goebel,  'Organography,'  ii,  131  :  Correns,  1899,  *n 
'  Untersuch.  Schwendener  ',  p.  385. 


tn  CryptogtMu  end 

to  it  as  far  as  it  will  go  (Correns,1  loc,  cit.  p,  3«/>i    The  tact  that  _ 

in  the  axis  may  be  associated  with  a  1  skied  apical  cell  in  cases  of 

dorsi  ventral i«  tars),  again  suggests  that  the  construction  is  a 

of  the  apex  of  the  shoot  as  a  whole,  and  the  details  of  the  twmnialfim  of 

the  apical  cell  are  really  subsidiary.    That  is  to  say,  instead  of  the  *eUei 

apical  cell  •  dominating '  the  orientation  of  the  leaf  members,  it  »  within  the 

control  of  some  more  obscure  m^fi^i*™  of  orientation  inherent  in  the 


plant-apex  The  serial  production  of  leaf-segments  by  watts  pa  rain!  to 
those  of  a  tetrahedral  cell  does  not  accurately  obtain,  ho»e»ef  much  n  has 
been  the  custom  to  regard  such  a  mechanism  as  'simple'.  The  (act  that 
each  cell-segment  from  the  apical  cell  gives  a  leaf-member  It  ff"r*rht*if  with 
the  close  approximation  of  the  resultant  laminae  while  young  in  a  terminal 


'  bud  '-aggregation,  affording  the  young  primordia 

aerial  environment ;  and  though  the  mechanism  in 

may  run  parallel  with  that  of  the  stems  of  certain  Ptcridophyta.  f 

obviously  no  identity  of  organization,  and  the  priority  of  Fibonacci 

ization  in  the  scheme  of  apical  growth  is  thus  traced  to  the 

migrant  land-flora. 

III.  Algae.    Examples  of  spiral  organization  in  r 


among  algae  are  restricted  to  the  Phaeophvceae,  in  so  Car  as  them 
alone  present  a  '  parenchymatous '  type  of  organization  in  their 
all  comparable  with  the  cellular  anatomy  of  higher  land-flora. 

The  thallus  of  Ckara,  based  on  a  corticated  nstmtet,  fecreasiaf  by  trass* 
verse  segmentation  of  an  apical  cell  bears  lateral  raanli  developed  at  a  lets? 
date  from  nodal  units ;  while  in  the  case  of  the  Florideae  the  Ifaafts*  is  ajsja 
always  more  or  less  based  on  a  filamentous  type  of  orgawzaboo.  oftta  eassfss* 
and  obscure,  but  apparently  never  attaining  the  stage  beyond  thai  of  dklotcsy. 
polychotomy,  or  indefinite  ramular  formations.  Types  in  whkfc  oW  tetoal 
ramuii  are  more  specialised,  or  acquire  definite  dorshroiral  Ofwstsdcsu  ast 
restricted  to  axes  of  a  single  filament  (cortkaied  types),  in  the  ssssaer  of  Casr* . 
e.  g.  in  Potysipkoma  the  ramuii  are  associated  wkh  conical  antes,  aad  hsnot  SB 


building  spirally,  one  member  at  a  time, 

of  a  similar  winding  spiral  ('staircase-effect '),  giving 

not  a  Fibonacci  system  (cf.  Chard  with  a  spiral  lystsni  of  oat 

and  Polysiphoma  with  four  cortical  cells  in  each  segment,  ghrtef  a  •  * 
Identical  phases  of  somatic  progression  are  associaaed  wtt  tt* 

segmentation  of  Phaeosporeae,  as  in  SMmbr*,  * 

lockmu,  giving  bilateral  or  balanced  (whorled) 

constructions. 

Among  the  more  highly  differentiated  Fucaceac  (in  the 
apical  segmentation  in  terms  of  a  singles-sided'  apical  cell 
again  as  the  limiting  expression  in  building  a  centric  shoot  bv  one  dossil 
cell  with  a  *  rotating '  nucleus :  the  massive  ramuii  are  not  dearly  t 
any  definite  segment  of  such  a  cell,  though  spaced  at  fairly  egejaj  sejereil 
Such  massive  somatic  outgrowths  may  acquire  a  certain  todMdesJ 
specialization,  as  expressed  in  limited  growth  and  special  farm  fee* 
their  own,  presenting  anything  from  an  elaborated 
to  a  simple  scale-like  growth  ;  all 
ramification  and  the  localization  of 


>  Correns  gives  the  angle  of  segmentation  wqefctd 
sequence  of  137°  30'  as  51°  31'  4*-     The  njvre  tejijei 

means  the  general  shape  of  the  j-skted  cell  of  a  Muss  apti,  md\ *tj 
so  far  only  of  academic  interest.    It  may  be  noticed  mtt  each  Sftftea, » 
by  curved  lines,  are  beyond  investigation. 


42        On  the  Interpretation  of  Phenomena  of  Phyllotaods. 

among  Cystoseircae  (in  the  widest  sense)  and  the  more  specialized 
types  of  Sargasseae  (including  Sargassum  and  Turbinaria). 

In  these  forms,  so  long  as  the  organization  of  the  shoot-system  remains 
centric,  Fibonacci  relations  arc  found  to  obtain  ;  though  the  presence  of  the 
mechanism  may  be  obscured  by  the  fact  that  growth  of  the  lateral  ramuli 
is  usually  more  open  ;  i.e.  not  telescoped  within  the  limit  of  a  terminal  bud 
which  may  be  sectioned  to  show  the  accurate  space-relations  of  the  adjacent 
members.  Further,  in  the  case  of  larger  frondose  systems,  the  accidents  of 
the  moving  medium  prohibit  any  precise  display  of  mutual  relations  or 
special  orientation  with  regard  to  a  fixed  light-position.  Hence  the  general 
plan  of  the  shoot-construction  is  less  noticeable;  though  in  a  few  form 
the  apical  construction  may  have  attracted  attention  (cf.  Landsburgia, 
Cystoseira).1 

In  this  way  one  is  entitled  to  assume  the  existence,  within  the  paren- 
chymatous  apex  of  the  Fucaceae,  of  a  structural  organization  apart  from 
that  of  the  cellular  segmentation  controlling  the  space-form  of  the  lateral 
members  of  a  centric  shoot-system,  and  giving  an  optimum  system  of 
distribution,  built  one  member  at  a  time,  and  possibly  of  distinct  benefit 
from  the  standpoint  of  the  maximum  illumination  of  the  members  of  a 
centric  shoot-system.  On  the  other  hand,  the  bilaterality  of  Halidrys  does 
not  give  '  minimum  superposition'  since  the  frond-systems  lie  extended  in  a 
plane  across  that  of  incident  light  in  a  flowing  medium,  or  held  vertically 
erect  by  means  of  the  pneumatocysts  indifferently.  Hence,  though  there 
are  apparently  two  alternative  constructions  in  the  sea,  one  is  not  entitled 
to  assume  that  they  have  been  produced  solely  in  response  to  problems  of 
maximum  and  minimum  illumination  ;  there  may  be  something  else  behind. 
On  the  other  hand  one  is  justified  in  concluding  that  Fibonacci  phyllo- 
taxis  was  initiated  in  the  sea,  without  any  necessary  connexion  with  the 
claims  of  insolation  in  land-vegetation.  Its  origin  is  to  be  looked  for  in  the 
sea  ;  although  in  transmigrant  land-vegetation  the  system  of  construction 
may  prove  valuable  under  the  new  conditions,  and  so  be  retained  as  one  of 
the  most  deeply  ingrained  construction-factors  of  the  leafy  shoot.  It  being 
so  far  clear  from  the  organization  of  the  Fucaceae  that  Fibonacci  relations 
are  older  phylogenetically  than  the  differentiation  of  the  'leaf  itself  as  a 
strict  morphological  entity  (as  defined  in  terms  of  subaerial  vegetation). 

In  Cystosrira  ericoides,  a  readily  available  and  indigenous  plant,  the  small 
subulate  ramuli  (3-5  mm.)  are  borne  in  more  definite  bud-aggregation  (1-2  mm.) 
at  the  apices  of  growing  shoots ;  though  in  the  strongest  of  these  the  tendency  of 
some  of  the  ultimate  ramuli  to  again  bifurcate  from  the  base,  or  to  branch  into 
two  or  three,  commonly  disturbs  the  effect  of  the  pattern.  Simpler  results  are 
obtained  \vith  smaller  laterals,  in  which  growth  is  less  active  and  the  ultimate 
ramuli  remain  prevailingly  single;  in  general  effect  much  resembling  the 
smaller  ramuli  of  such  a  plant  as  Arautaria  excelsa  on  a  smaller  scale;  the 
ramuli  presenting  no  special  trace  of  dorsiventrality  and  ending  in  a  sharp  point. 
In  such  a  shoot  a  view  of  the  terminal  bud  from  above  shows  obvious  Fibonacci 
symmetry ;  and  though  angular  measurements  cannot  be  given  with  any  degree 
of  accuracy,  the  general  resemblance  of  the  pattern  to  the  Fibonacci  systems  of 
higher  plants  is  very  striking.  In  the  bud  figured  (Fig.  XVII)  10  members  in 
different  stages  of  growth  may  be  distinguished,  and  number  6  falls  in  the  gap 
between  i  and  3 ;  there  is  thus  no  question  of  a  '  f '  divergence,  though  9  is 
nearer  superposition,  as  if  referable  to  a  '  f  '  system.  The  divergence-angles 
as  measured  in  the  figure  are  : — 132°,  127°,  142°,  154°,  127°,  130°  for  members 
1-8,  giving  an  average  of  135-3°,  again  approximately  a  'f  divergence  of 


1  Oltmanns  (1904),  'Algae  ',  p.  505.     Vaillant  (1883),  '  Flora  and  Fauna,  Gulf 
of  Naples ' :  Cysloscira.     Grubler  (1896),  Bibliot.  BoL,  Heft  38,  Taf.  7  :  Landsburgia. 


Phyllotaxis-phenomtna  in  Cr#Ugumt  *md  TktlbfikyU.    43 


Schimper,  bat  well 
no  accuracy  can  be  guaranteed  in 
(i)Onlra  few  members  ire  available ;  the 
more  are  in  right    (a)  There  is  no 


range  of  individual  error  is  again  lane ;  e.g.  i»7*-i§4*, 
the  phenomena  closely  approximate  those  of  the  P^fyi 

The  critical  value  of  this  Fucoid  apex  centres  to  the  feet  that  while  tike 
centric  axis  is  dominated  by  a  3 -tided  apical  cell,  cutting  off  segments  to 
three  series,  much  as  in  the  Mo**,  the  phyllotaxi*  again,  even  if  regarded  a* 
based  on  some  derivation  of  cell-segments  ol  the  apical, presents  undoubted 
'  compensation '  in  the  organization  of  the  apex  as  a  whole ;  and  the 
primordia  assume  the  Fibonacci  orientation  by  a 
involving  growth-distribution  within  the  young  sh 
there  is  no  direct  proof  of  segmental  origin,  the  equal  spacing  of  the  • 
may  well  suggest  that  there  must  be  some  correlation  between  the 
and  the  three  rows  of  segments  diverging  from  the  apical  cell ;  but 
ever  it  may  be  is  immaterial  to  the  general  organization.  This  may  seem 
somewhat  arbitrary  so  far  as  Cystoseira  tricouks  is  concerned,  but  more 
conclusive  evidence  of  such  secondary '  compensation'  is  afforded  by  the 
associated  example  of  Haluirys.  Since  in  the  latter,  with  an  ide 
of  apical  segmentation  (Oltmanns,  1904,  p.  510)  the  lateral 
*  compensated  '  by  undoubted  secondary  growth -processes  to  rive  a 
tion  of  obvious  bilaterality,  and  the  laterals  (all  into  two  strict  lateral 
stichies  as  a  pinnate  frondose  type.  That  is  to  say,  if  Halidrjt  can  c 
a  system  derived  from  a  segmenting  3-sided  apical  cell  to  bilatcralrty.  so 
may  Cystoseira  change  the  same  cell-organization  to  Fibonacci  symmetry ; 
since,  assuming  the  3-sided  cell  would  give  3  orthostichks  at  iao*.  the 
change  from  120°  to  the  180°  of  bilaterality*  is  much  greater  than  the 
compensatory  correction  of  iao°  to  137}°  required  for  the  Fib 
struction  ;  especially  as  the  individual  error  of  one  ramulus  has 
to  range  to  127°  in  the  adult. 

the  end,  again,  as  in  the  Moss,  one  is  inevitably  led  to 
sion  that  the  orientation  of  the  lateral  ramuli  is  not  only  < 
of  the  shoot  as  a  whole,  but  that  the  details  of  apical- 
subservient  to  this  function,  whether  expressed  in  primary  centric 
relations  or  in  secondary  bilateral  symmetry.     It  may  be  noted  that 
with  the  best  intentions  one  cannot  fairly  call  such  ramuli '  leaves', 
as  elaborated  frond-systems  of  species  of  Sarfassttm,  or  reduced  to 
scales  as  in  Cystoseira  cricoides  ;  dorsivcntrality  is  wholl; 


bilaterality  may  be  obscure,  as  such  ramifications  attain  a 

habit.     But  it  now  follows  that  the  Fibonacci  system  of  mtflBliUWMl  If 

older  than  the  attainment  of  the  'leaf  as  understood  in  sob-aerial 

tion,  and  is  a  phenomenon  of  response  to  conditions  of  marine  eyfto 

so  far  beyond  the  horizon  of  all  land-vegetation,  and  only  surviving  to 

latter  as  a  part  of  the  inherited  equipment  of  the  »ea ;  though  furthr 

izcd  and  utilized  as  it  proves  efficient  under  new  conditions  of 

progression. 


44       On  the  Interpretation  of  Phenomena  of  Phyllotaxis. 

VIII 
QUINQUELOCULINA 

ONCE  it  becomes  clear  that  Fibonacci  symmetry  in  the  production  of 
lateral  appendages  of  a  main  axis,  in  acropetal,  spiral  succession  is  a  pheno- 
menon general  among  more  massive  marine  Algae,  as  the  response  to  a 
condition  of  submarine  environment,  the  story  of  phyl lota xis  takes  a  new 
turn  ;  since  it  is  obvious  that  the  presentation  of  Fibonacci  factors  in  the 
case  of  subacrial  vegetation  can  be  only  interpreted  as  the  retention  of  an 
ancient  mechanism,  adapted  now,  it  may  be,  to  the  insistence  of  problems 
of  insolation  on  land,  but  of  undoubted  primary  relation  to  a  wholly  different 
set  of  physical  conditions  ;  yet  so  deeply  impressed  that  it  is  retained  more 
or  less  clearly  for  all  time  in  transmigrant  vegetation.  Similarly  it  now 
becomes  of  interest  to  look  in  other  directions  for  any  comparable  illustra- 
tion of  the  same  general  principles,  which  may  afford  a  guide  to  the  more 
deep-seated  and  fundamental  factor  involved.  Such  a  clue  may  be  possibly 
found  in  a  race  of  entirely  different  organism,  the  Foraminifera,  as  a  hetero- 
trophic,  holozoic,  animal  phylum  of  Protozoa,  without  chloroplasts,  and 
with  no  photosynthetic  problems,  yet  building  in  terms  of  a  unicellular  soma 
an  organization  which  presents  clear  and  definite  factors  of  Fibonacci 
symmetry. 

The  atteniion  of  botanists  to  these  remarkable  little  organisms,  both  living 
and  fossil,  was  first  drawn  by  Van  Iterson  (Delft,  1907),  and  many  figures  from 
zoological  and  geological  works  of  Munier-Chalmas  and  Schlumberger  are  given 
in  his  work  on  Blattstellungen.1  Obvious  Fibonacci  constructions  are  presented 
in  such  types  as  Quinqucloctdina  vulgar  is,  Triloculina  rotunda,  Periloculina  Rain- 
courti,  PtnUllina  Douvillei?  Forms  are  also  found  in  which  juvenile  centric 
organization  grades  into  bilaterality  in  the  adult  (Heterillina  Guespcllensis)?  as 
other  types  express  strict  bilaterality  from  the  first.  Other  variants  may  be 
symmetrically  2-spired,  or  even  3-spired  in  the  manner  of  the  Polytrichum  bundle- 
traces  ;  cf.  Spiroloculina,  Idalina  antiqua,  and  Trillina  Hmvchini,  Schlumberger 
(1893).  The  Fibonacci  construction  among  such  types  is  but  one  out  of  many 
possible  systems ;  though  generally  accepted  as  the  optimum  and  so  far  the 
highest  expressions  of  somatic  specialization. 

It  remains  to  analyse  the  factors  of  the  growth  of  such  organism,  and 
to  discuss  the  special  value  of  the  phenomena  observed.  Preparations  from 
geological  or  zoological  works,  or  from  sections  of  decalcified  material,  are 
less  satisfactory,  as  being  doubtfully  expressed  in  exact  measurement.  Yet 
Iterson  (p.  311,  loc.  cit.)  gives,  for  Pentellina  Douvillci,  a  series  of  10  suc- 
cessive divergence-angles,  as  143°,  140°,  127°,  152°,  127°,  135°,  142°,  128°, 
147°,  134°.  The  extreme  range  from  127°  to  152°  again  gives  a  wide 
latitude  ;  but  so  far  the  average  works  out  as  137-5°,  which  leaves  the 
matter  sufficiently  suggestive  as  a  phenomenon  undoubtedly  expressed  in 
terms  of  the  same  general  Fibonacci  category,  and  requiring  further  exam- 
ination. An  illustration  of  a  transverse  section  of  Quinqueloculina  seminu- 
lum,  figured  by  Worth  (1907)  in  the  Journ.  M.  B.  A.,  appeared  so  beauti- 
fully cut  and  symmetrical  that  it  might  be  taken  as  a  type  (Fig.  XIII). 


1  Van  Iterson,  jun.  (1907),  '  Mathematische    und   mikroskopisch-anatomische 
Studien  fiber  Blattstellungen ',  p.  299. 
1  Van  Iterson,  loc.  cit.,  p.  306. 
8  Cf.  Biloculina  dtprasa,  Minchin,  (1912)  Protozoa,  p.  233. 


f  h ,   '.,,  • 

Foramu^ZToflbMthsc  habit,  With  calcareous  lest 
common  in  every  latitude  from  arctic  seas  to  the 
water  to  3,000  fathoms  (as  tests  only).    It  is 
Eocene,  and  in  all  deposits  since  (' 


The  special  nature  of  the 
from  external  view;  the  organism 
chambers,  convex  peripherally,  with  oral 
podia.    A  transverse  section  half-way  op 'the 
The  spucfanm  figured  was  just  over  i  nun. 
peripherally,  showed  about  jo  chambers,  the  majority 
urred  in  a  pebble  of  Eocene  hmestone,  dredgSd  38 
Eddystone,  and  belongs  to  a  collection  of  the  rocks  of  the  district.    Tie  sndt 


re.     A  large  camera  lucida  drawing,  under  the 
diameter,  showed  up  the  minor  errors  of  the  construction,  and 

construction.    The  organism  apparently  sank  with  vertical 
pebble-section  followed  a  horisontal  plane  with 
The  figure  clearly  shows  that  while  the 
founded  on  the  '  quasi-square ',  with  minor  icnlptnral 
each  is  theoretically  circular,  being  equally  clearly  a  r 
quasi-circlc,  and  bilaterally  symmetrical  in  the  plane  of  section  with 
a  radius  of  the  system  as  indicated  by  the  dotted  lines  used  for  BACK 
divergence-angles.    Extreme  flartr^fog  obtains  in  the 


foot  in 

daw 


but  at  the  optimum  region  of  the  construction  (No*,  t»is>  the 
quasi-ftquares  and  quasi-circlct  is  the  most  remarkable  and  most 
of  the  system.    It  may  be  also  noted  that  the  genen  " 
rowed '  Trikculina  and  '  five-rowed '  Q*i*fmt£rmtm*t  is 
the  amount  of  tangential  extension  in  the  units,  and  hi 
Fibonacci  angle.    Extreme  tangential  extension  leads  on  to  the 

ility  only,  as  in  Biloculina,  Sf>trondtma,    To  any  botanist  the 
good  example  of  a  •  bud  '-section. 


of 

Sift 


Examination  of  the  outermost  chambers  shows  minor 
some  of  these  encroach  beyond  their  legitimate  territory, 
form  varies  somewhat  towards  the  periphery,  with 
luring  of  the  outer  margin  of  the  test,  a-,  3-,  4-. 
members  appear  more  tangcntially  extended  or  flattened.  These  detala 
express  the  general  plan  of  growth  in  adding  new  chambers  to  the  test  MT 
at  u  //;//<-,  on  a  pre-existing  base ;  general  accuracy  " 
remarkable  perfection.  The  *  con  tact -parastlchics §  c 
a  :  3  system  ;  the  4  genetic  spiral '  can  be  traced  on 
in  the  usual  way,  and  in  this  case  it  winds '  Right '  from  the 
The  following  divergences  were  measured  on  the  plan,  with 
within  the  error  of  observation,  just  as  in  the 
apex : — 


of  list 


1-3 

'-3 
3-4 


•M' 


9-10 

10-11 

it-it 


: 
17-1* 


1  Lankestcr  (1903),  'Treatise  on  Zoology.'  p.  SS,  do 
chambers  on  5  radii,  as  it  is  formed  by  a  rotation  of  |;  d  TVs! 

(bilateral). 


46        On  tJie  Interpretation  of  Phettomena  of  Phyllotaxis. 

The  range  is  again  considerable  (max.  i56°-min.  129°),  and  the  difficulty  of 
accurate  measurement  is  increased  by  the  fact  that  there  is  no  central 
point,  as  in  the  case  of  the  central  protoxylem  of  a  leafy  shoot :  measure- 
ments are  thus  approximated.  Even  where  accuracy  was  most  attainable 
measurements  did  not  come  very  exactly  on  the  137?°  mark,  but  rather 
average  140°  ;  but  of  the  beauty  of  the  system  as  a  whole  there  can  be  no 
question.  Taking  averages  of  5  successive  members  at  a  time,  as  a  full 
cycle,  to  eliminate  errors  of  centring  (as  in  Semper  viiwn),  successive 
averages  range  from  142°  to  144-5°:  the  average  of  the  entire  sequence 
(18  members)  was  142-8°;  a  figure  at  first  sight  far  removed  from  I37?0.1 

Perhaps  the  most  striking  feature  of  the  construction  is  the  exact  radial 
orientation  and  consequent  *  isophylly '  of  the  loculi  of  the  chambers,  recall- 
ing the  strict  radial  orientation  of  the  derivatives  of  quasi-circles  in  an 
orthogonal  construction  ;  such  a  detail  was  quite  unexpected,  and  is  un- 
doubtedly equally  confirmatory  of  the  fundamental  mechanism  behind  the 
whole  construction.  There  is  thus  no  doubt  whatever  of  the  presentation 
in  such  an  organism  of  perfectly  normal  Fibonacci  symmetry  in  building  the 
somatic  chambers.  The  only  questions  are  (i)  How  is  it  done,  and  what  is 
the  mechanism?  (2)  Why  is  it  done?  These  problems  being  bound  up 
with  the  organization  of  the  Foraminifera  as  a  class.8 

The  Foraminifera  are  to  be  regarded  as  a  group  of  Protista,  derived  from 
an  ancient  pelagic  flagellated  series,  residual  traces  of  which  are  still  retained  in 
reproductive  phases,  as  simple  isokont  or  monokont  (Pentroplis)  zoVds  of  the 
most  elementary  type.  But  with  the  assumption  of  a  benthic  and  holozoic 
habit,  nutrition  is  effected  by  means  of  a  pseudopodial  net  catching  plankton- 
rain  of  small  dimensions  (diatoms,  &c.),  the  flagellated  stage  being  suppressed  in 
the  adult ;  while  the  periphery  of  the  soma  is  enclosed  by  a  precipitated  calcified 
test  leaving  an  oral  aperture  (or  others)  for  the  extrusion  of  the  net.  In  the  case 
of  regressive  plankton-forms  the  test  must  act  as  a  gas-holder  for  purposes  of 
flotation,  thus  again  replacing  necessity  forflagellar  action  (Globigerina).  Where 
such  food-supply  is  abundant  and  a  substratum  is  available,  a  more  benthic  habit 
affords  a  satisfactory  solution  of  the  problem  of  nutrition,  and  such  organism 
may  attain  considerable  dimensions.  But  once  confined  by  a  rigid  test,  increas- 
ing growth  is  expressed  in  the  production  of  new  regions  of  similar  form, 
similarly  ultimately  fixed  by  the  calcareous  deposit ;  the  newer  chamber  being 
again  usually  larger  than  the  preceding,  though  approximately  of  the  same  general 

1  By  actual  measurement  the  divergence  angle  is  obviously  fairly  correct  for  the 
Archimedean  angle  144°,  and  would  be  regarded  as  an  unsatisfactory  approximation 
to  the  theoretical  log-spiral  angle  of  138°  27'  42"  (R.P.M.L.,  p.  340).    Examination  of 
the  figure,  however,  shows  that  after  the  first  few  members  the  radial  depth  of  the 
unit  remains  practically  constant,  and  growth  is  adjusted  in  the  tangential  component. 
The  log-spiral  system  implying  uniform  expansion  of '  similar  figures '  is  not  main- 
tained, and  the  Archimedean  construction  is  the  result  of  a  secondary  factor. 

It  is  also  interesting  to  compare  the  range  of  error  in  the  case  of  the  individual 
unit,  as  an  amplitude  of  oscillation  about  the  mean  of  137-5°,  in  : — 

Quinqwloculina    .     .     .   1 56°    -  1 29°. 

Pentdlina  (Iterson)  .     .   152°    -127°. 

Cysiosa'ra  (¥\g.  XVII)  .   154°   -127°. 

Polytrichum  (Fig.  XVI)  152-5°-  127.5°. 

i.e.  as  much  as  10  degrees  either  way :  such  a  range  of  error  at  once  eliminates  all 
question  of  the  necessity  of  working  accurately  even  to  degrees ;  as  it  also  indicates 
the  community  of  'design*  or  'response '  in  a  wide  range  of  organism  in  their  relation 
to  Fibonacci  symmetry. 

2  Minchin  (1912),  'Protozoa',  p.  231. 
Lankester  (1903),  'Treatise  on  Zoology',  i,  p.  47. 
Winter  (1907),  Pcneroplis,  A.P.K.,  x,  p.  16. 
Rhumbler  (1903),  A.P.K.,  p.  181 :  1902,  p.  252. 


type.    PJfcfgm^pssof  Fonmhjfuift  stay  be  •iifmnfili 

every  pc*dbifcT_of_spatkl  extension  as  wS'as  pitsen*n?a  wide  rang,  of  • 


•naked*  forms  grade  into  other 
organism.    As  different  phsset  of 

growth  in  a  linear  series,  as  •«rtimt4'*n  to  one 
(s)  growth  in  a  plane  (twc-diinensional)  giving  aksmsi 
skUoftheorigir^tc»t,^/*W/«;(3)kxma^ofthtntw 
the  same  side,  with  a  bias,  giving  a 

1  *_  —       ^2a>sV     nsV ^     —  — — +> _/    _ 

SSBJSh   wSfli   BM    OOBSpMntSSl   "     i 

for  three-dimensional  growth,  from  the 


f  i  • 

tion  of  centric  symmetrir,  and  exposure  of  minlniiai  stvfaot  of  inc.  »  dasfftf 
the  Fibonacci  fflftttnictioPt  so  V4>g  as  only  flttf  new  chamber  fitt  be  bofll  at 
onetime.  With  this  proviso,  thePibooacd  angle  of  137  §•  si  the  aw** 
solution  of  the  problem,  and  it  is  interesting  to  find  that  it  mar  be  so 

• 


approximated  ;  while  forms  presenting  such  a  conjunction  art  so  sVof 
higher  grade.1 

mechanism  of  the  process  appears  fairly 

unit  restricted  to  a  definite  form, 

latter  must  necessarily  fall  on  one 

linear  series);  and  if  the  second  unit  remains  to 

the  same  central  control,  the  original  centric  BQSJMbrtosi  of  the 


appears  Curly  snoesthe;  givtn  a  olaMnflc 
building  a  second  unit  of  rinsflar  tort*,  das 
tMf  or  the  ^***^^  (osssnss«  aW  C^M  tt.  sW 

^**     »«^w    ^T««S^BV    m^^ss^ssvsinsBnm    ^nssji    %^sjsvnj    ^M    sjssW 


only  regained  as  the  third  unit  makes  good  toe  balance  of  tin  muss  ,  cttbar 

falling  at  180°  from  the  second,  giving  the  bilateral  or  two  riasnaJonsI  conoV 


tion,  or  balancing  in  the  optimum  three-dimensional  position  at  i  jjj*.  and  so  on 

in  successive  units.     From  such  a  standpoint,  the  •  balance  *  appears  as 

version  of  the  general  principle  of  growth  in 

surface-tension  in  a  fluid  medium ;  and  the  fluid 

streaming  automatically  to  the  next  position  oi 

spheroidal  phase,  and  being  ultimately  fixed  by  its  deposit  of 

precipitated  excreta  of  calcium  carbonate.     In  this  way  the  •tAsaboo  of 

Fibonacci  angle  appears  as  a  fundamental  property  of  gi 

fined  within  a  rigid  boundary,  under  the  control  of  a 


version  of  the  general  principle  of  growth  in  conflict  with  the  drssandi  of 

;  and  the  fluid  cytoplasm  may  be  vfanaftasd  i 
streaming  automatically  to  the  next  position  of '  balance '.  as  inherited  fcom 


which  necessitates  the  building  of  new  extensions  of  the  sots*  *v  *  * 
Of  the  possible  solutions  of  this  problem  presented  to 
Foramintfera,  the  Fibonacci  system  is  clearly  the  bast,  s 
unit  which  alone  retains  on  the  whole  its 
degree ;  Le.  so  long  as  it  follows  the 


the  same  degree  of  approximation  to  the 

such  automatic  '  balancing ',  again,  may  be 

so  far  unconscious,  and  beyond  the  immedl 

is  no  need  to  postulate  a  conscious  '  thinking  out ',  or 

of  137°  30'  20.9",  a  factor  dearly  beyond  the  capacity  of  any 

plant.    But  just  as  a  man  on  a  bicycle,  when  amosssricsly  balanced  fa  at  an 


no  necessity  to  demand,  or  to  be  expected  to 
angle  in  any  given  case,'    Given  an  inherited,  or 


*des  of  OrMotito,  by  balding 

as  « similar  figures',  many  at  one  time,  may  Assent  patterns  at 
comparable  with  the  capituhtm  of  CunTjiosilsSi  thongh  lendtog 
spiral  parastkhies '.     Rhumbler,  A.P.K.,  1901,  p»  149. 

1  This  also  explains  at  once  why  to  the  case  of  h%ner  plsntoany  pasr  « 
Fibonacci  numbers  from  a :  3  onward  wffl  give  equally  satisfactory  rasssX  since  to 
this  ratio  only  is  the  error  from  137^  as  moch  as  one  dojrm    (R-PJtU  n. 


48        On  the  Interpretation  of  Phenomena  of  Phytto taxis. 

and  symmetrical  growth-distribution,  the  angle  follows;  approximated  on  the 
whole,  in  the  average  of  successive  members  around  the  centric  field  of  growth, 
though  with  a  possibly  wide  margin  in  the  case  of  individual  units,  as  these  may 
vary  in  accordance  with  the  interposition  of  a  time-factor.1  It  further  begins  to 
be  clear  that  what  holds  for  the  fluid  plasma  of  a  unicellular  soma  will  equally 
hold  for  the  plasma  of  a  multicellular  soma,  similarly  inheriting  centric  symmetry  ; 
just  as  the  approximate  cylindrical  form  of  a  tree-trunk  is  but  the  reflect i 
the  massive  soma  of  the  primary  claims  of  surface-tension,  which  gave  centric 
form  to  the  first  passively  suspended  plankton-cells. 

The  essential  point  to  note  is  the  almost  absurd  manner  in  which  the 
Fibonacci  pattern  of  Quitujueloculina  resembles  that  of  a  plant-apex,  although 
the  two  constructions  are  worked  out  in  diametrically  opposite  terms.  Thus, 
Quinqueloculina  builds  one  chamber  at  a  time,  placed  on  the  periphery  of 
the  older  test  which  is  itself  no  longer  growing  :  the  effect  of '  growth  '  and 
consequent  log-spiral  constructions  will  be  merely  the  expression  of  the 
fact  that  successive  units  are  similar  figures ;  only  differing  in  progressively 
increased  dimensions.  While  in  the  case  of  the  plant-apex,  the  new  mem- 
bers are  added  one  at  a  time  internally^  at  the  centre  of  construction  of 
a  growing  system  which  is  expanding  throughout  its  entire  mass  ;  the  log- 
spiral  effect  being  again  the  result  of  the  units  remaining  similar  figures  as 
they  all  continue  to  grow  in  graded  sequence. 

Only  one  factor  is  common  to  both, — the  building  of  new  units  one  at  a 
time, — and  it  thus  appears  that  this  is  the  essential  factor  behind  all  such 
presentation  of  Fibonacci  relations,  to  all  time.  Fibonacci  symmetry  is  not 
only  older  than  the  'leaf;  it  traces  back  beyond  the  differentiation  of  stem 
and  leaf,  and  is  one  of  the  most  fundamental  properties  of  living  organism. 
Established  in  the  working  mechanism  of  a  'growing-point',  itself  the 
elaboration  of  the  axes  of  benthic  seaweeds,  it  is  deeply  ingrained  in  the 
constitution  of  all  subaerial  vegetation,  and  may  be  in  turn  one  of  the  last 
factors  to  be  lost  in  the  general  deterioration  of  apical  mechanism.  This 
applies  with  special  emphasis  to  the  case  of  floral  shoots  in  which  the 
mechanism  of  leaf- production  fails  as  the  flower  attains  a  more  finite  and 
minimized  organization. 

As  a  secondary  feature  in  which  Quinqueloculina  again  conies  into 
line  with  the  spiral  bud-systems  of  higher  land-plants,  may  be  noted  the 
isomorphy  (isophylly)  of  the  respective  units,  as  they  present  strict  bilateral 
orientation  with  regard  to  radii  of  the  centric  system.  Since  such  a  property 
is  apparently  restricted  to  spiral  constructions  in  terms  of  orthogonally 
intersecting  log-spiral  curves  2  (or  legitimate  derivatives  from  such  a  system 
by  the  addition  of  a  new  factor),  it  follows  that  in  both  cases  the  distribu- 
tion of  growth  activity  may  be  visualized  as  following  the  lines  of  physical 
forces  acting  in  orthogonally  intersecting  planes,  a  factor  common  to  other 
examples  of  distribution  of  '  equipotential ',  and  so  far  a  physical  property 
of  matter.8 

By  showing  that  a  mechanism  of  strict  Fibonnaci  symmetry  is  in 
practicable  working  order,  even  in  a  holozoic  race  of  marine  Protista,  it  is 
not  intended  to  encourage  any  ridiculous  idea  that  marine  Algae  must 
hence  present  some  4  affinity '  in  these  respects,  or  even  may  have  had 
a  common  origin  in  some  autotrophic  organism  similarly  exhibiting  such 
a  growth -median  ism.  Though  it  is  quite  obvious  that  countless  races  of 
Protista  and  low-grade  Algae  may  have  existed,  and  have  been  lost  in  the 

1  Hence  in  higher  ratios  accuracy  will  tend  to  increase  as  a  larger  number 
of  units  (typically  a  full  contact-cycle)  may  be  initiated  practically  simultaneously. 
1  R.P.M.L.,  loc.  cit.,  p.  241. 
3  Ibid.,  p.  230. 


•   - 


49 

remote  past,  the  fact  remains,  that 
Fibonnaci  <,yinmctry  is  first  traced  in 

hacophyceae,  as  it  must  also  have  been 
chymatous  massive  marine  Algae  from  which  the  higher  vegetation  e7  tke 
land  (Bryophyta,  Pteridophvta,  Phanerogams)  has  been  derived »  Tke 
mechanism,  that  is  to  say,  has  been  acquired  polyphylcticaily  at  a  coo 
sequence  of  the  adaptation  of  a  special  mode  of  growth  fa  verv 
groups,  and  it  remains  to  analyse  its 
Qitinquelocnlina  as  a  guide,  the  production  of 
represents  a  local  extension  of  the  somat 
the  calcareous  precipitate  of  the  chamber-wall,  and  is  so 
increasing  the  surface-area  of  the  body,  in  opposition  to 

h  tends  tu  pull  the  organism  into  a  state  of  •^•Hfsj 
;ie  essential  clue  to  the  situation.    Th< 
case  of  the  original  primary  conflict  between 
and  plasmatic  growth,  which  is  the 

nn  of  elementary  organism,  bolophytic  or'  holoiok, 
special  solution  of  this  problem.  The  case  of  the  loid  with  a 
extension,  but  no  further  development,  U  the  first  umpic  a 
discussed.  The  problem  of  the  centric  Diatom  is  mock  more 
and  affords  the  clue  to  a  vast  range  of  special  samatk 
plant-nature.  Beginning  with  a  surface-tension  sphere,  the 
of  the  soma  to  a  circular  disk  expresses  the  restriction  of  growth  fa  as* 
dimension,  but  not  in  the  other  two.  The  meaning  of  tke  rirnress  is 
sufficiently  clear ;  the  centric  Diatom,  the  discoid  chioropUst  of  tke  asju>- 
trophic  land-plant,  and  the  discoid  blood-corpuscle  of  higher 
express  the  retention  of  this  simple  phase  of  organism  •orhhsf  to 
of  surface-action.  Beyond  this  stage*  growth-thrusts,  symmetrically  spaced 
with  geometrically  balanced  accuracy  (as  a,  Atofasfc.  3. 
4,  5,  6,  8,  10,  &c.,  of  many  Diatom-forms)  afford  a 
solution  of  the  problem,  fixed  by  a  later  deposit 
almost  mathematical  accuracy.  The  points  to  note  arc,— (i»  the 
thrusts  are  equally  distributed,  more  than  one  at  a  time; 
initiated  once  for  all :  there  is  no  possibility  of  further  addition,  and  «*V 
system  is  so  far  dosed  \  (3)  they  are  orientated  fa  one  pkat  as  a  two- 
dimensional  phenomenon. 

From  such  a  standpoint  may  be  considered  tke  eyaisjnlt  of  one  suck 
thrust  at  a  t  :»  the  possibility  of  icugUtion  ;  this  is  tke  case  of  tkr 

Foraminifcr  of  the  Quinqutlocuhna  type.    The  factors  being 
sional   as,— (i)   a   longitudinal   axis  which   is   relatively 

Kibonacci  symmetry  in  th«-  other  two  dimensions  in  a  plane  at  right 
angles  to  the  former. 

That  is  to  say,  the  Fibonwci  symmetry  it  a 
and  tke  solution  of  tke  problem  of  m  growtk^kmst  in  o»o*i1im  * 
surface-tension  (though  still  working  in  terms  of  local 
now  presents  tke  additional  possibility  of  being  rep*** 
a  growing  pattern  in  an  asymmetrical  or  *  spiral*  JSSWSJP 

>n  of  symmetry,  combined  with  tke  complete  ifjstiktfan  of 
growth,  which  is  the  essential  factor  dommathw  tke  jJilfcmisnmVlH  of 
Diatom-soma.  and  is  also  responsible  for  the 
otherwise    remarkably    kilffafotfiyk' 
distribution  of  growth-form  in  dimensions  at  rigfct 


1  Church  (1919)^ ' 
urch(i9i 


Thakssiophyta*.  ^  9* 


VI 


5O       On  the  Interpretation  of  Phenomena  of  P/iy/. 

expresses  the  fundamental  claim  of  the  general  principles  of  orthogonal 

t  ruction  in  living  organism. 

Any  race  of  organism  which  thus  combines  growth  in  one  longitudinal 
direction,  as  distinguished  from  the  other  two  spatial  dimensions,  with 
growth  in  the  latter  at  right  angles  to  the  first,  distributed  one  member 
at  a  time,  must  inevitably,  as  the  optimum  solution  of  the  surface-tension 
problem,  work  out  a  Fibonacci  system  in  the  plane  transverse  to  the 
longitudinal  axis.  Because  a  benthic  plant  develops  a  longitudinal 
as  its  first  asset  in  protobcnthon  of  the  sea,  it  follows  that  as  soon  as  it 
builds  lateral  ramuli,  one  at  a  time,  in  rhythmic  sequence,  these  should 
follow  the  Fibonacci  rule.  Observations  of  the  somatic  organization  of 
Phaeophyceae  and  Bryophyta,  show  that  this  has  been  tin  case,  It  is 
now  clear  why  phyllotaxis-phcnomcna  are  to  be  considered  solely  with 
reference  to  the  transverse  component  of  the  apical  growth-activ 
The  longitudinal  extension  of  axial  growth,  which  gives  the  'spiral' 
appearance,  is  a  compound  factor  which  must  be  analysed  into  its 
orthogonal  components.  The  higher  plant  retains  in  its  organization  this 
inherited  response  to  asymmetrical  growth,  just  as  the  cross-section  of  it> 
main  axis  normally  retains  the  transverse  component  of  the  older  surface- 
tension  sphere. 

The  case  of  a  single  growth-thrust  from  the  unmodified  sphere  itself, 
followed  by  others  at  different  points  on  the  sphere-surface,  is  more  complex, 
but  is  not  known  to  occur  in  the  plant-kingdom  ;  it  could  only  obtain 
in  a  suspended  plankton-organism ;  hence  its  geometry  need  not  be  discussed. 
But  simultaneous  equal  growth-thrusts,  in  three-dimensional  distribution, 
afford  the  basis  of  the  geometrical  relations  of  the  remarkable  holozoic 
group  of  Radiolarian  Protista.  In  the  case  of  the  holophytic  plant  this 
is  again  ruled  out  of  the  question  ;  though  whorled,  symmetrical,  and  equal 
growth -thrusts  are  beautifully  expressed  in  the  apical  construction  of 
coenocytic  Siphoneae  (Neomcris,  Dasycladus,  Acctabularia\  without  any 
necessary  relation  to  antecedent  spiral  phases. 

Fibonnacci  phyllotaxis,  as  a  phase  of  plant-symmetry,  thus  reduces 
to  a  condition  of  centric,  axial,  growth-extension,  combined  with  the 
out-thrust  in  rhythmic  sequence  of  somatic  protrusions  ('  ramuli ')  in  the 
transverse  plane.  To  maintain  the  older  inherent  centric  organization, 
the  Fibonacci  angle,  137°  30'  28",  must  be  approximated  every  time,  though 
the  range  of  error  may  be  considerable  in  the  individual  units.  Actual 
measurements  show  that  the  general  plan  keeps  very  fairly  adjusted  in 
such  divergent  types  as  Quinqneloculina,  Cystoseira,  Polytrichnm,  and 
o  cittpervi  vuin» 


IX 

PHYLETIC   PROGRESSION    IN    PHYLLOTAXIS- 

MECHANISM 

IN  dealing  with  such  a  complex  range  of  phenomena  as  that  afforded 
by  the  phyllotaxis-relations  of  higher  land-flora,  one  can  only  approach  the 
subject  from  the  standpoint  of  the  land-plant  itself,  in  its  modern  aspect ; 
knowing  quite  well  that  such  vegetation  was  never  created  directly  for  the 
position  it  now  holds,  but  is  the  outcome  of  a  long  series  of  progressive 
adaptation ;  so  that  the  modern  equipment  of  a  land-plant,  though  often 
apparently  admirably  suited  for  the  necessities  of  its  present  environment, 


PkyUHc  Progression  in  Pkytl*t*jris*mak*xi$m. 


must  nevertheless  have  been  initiated  in  response  to  a 

different  stimuli.     Kvery  organism  that  exists  at  this  i 

history  has  been  *  adapted ',  possibly,  over  and 

of  scientific  discut&ion  will  be  to  trace  such  adaptations  to  their  inal 

t -sighted  sub-aerial  tcleolog  may  attempt  to  find  a 

in  the  more  immediate  claims  of  present 
any  satisfactory  conclusion,  or  meet  with  wi 

In  analysing  such  a  maie  of  component  factors  a*  ririiaHiJ  by  the 

ion  of  ohy llotaxis-constructions  to  external  conditions,  it  b  first 
to   /raff  batkward,  and    to  deduce  from  the   mau  <*tr 

i  which  may  be  quite  imperfect)  what  can  be  so  Cur  definitely  established  as 
tending  to  the  ultimate  factors  of  causation,  step  by  step;  »  that  what  fc» 
established  from  deductions  afforded  by  the 

is  left  still  undecided,  may  be  dearly 

It  has  been  suff.  monstrated  that  all 

tions,  whatever,  in  higher  plants,  as  phenomena  of 
can  be  readily  discussed,  scheduled,  and  figured,  m 
curves  (the  •  contact -parastkhies '),  and  in  no  other  way. 
applies  equally  to  spiral',  •  whorletl ',  •  bijugate '.  and 
as  also  to  cases  of  •  rising  *  and  •  falling '  phyUotaxts ; 
with  phases  of  ultimate  deterioration,  which  present  no 
and  thus  can  be  scarcely  considered  as  rhythmic,  thoufh  \ 
may  often  remain.1  There  is  no  reason  to  suppose  that 
of  rhythm  is  the  primitive  case  for  any  high-grade  c 
organism  i>  presumably  rhythmic,  and  the  primitive  laws  of  'form*  at* 
not  based  so  much  on  the  irregular  holozoic  amoeba,  as  on  the  radiate 
mctry  of  the  surface-tension  sphere  of  still  earlier  autotrophk  organism. 
It  may  be  fairly  concluded  that  such  curves  afford  the  best  doe  to  the 
mecli 

II.  The  most  striking  and  most  constant  feature  of  all 
ms  is  the  fact  of  their  '  concentrated '  packing,  in 

known  as  the  'quincuncial' 

the  custom  of  planting  in  diagonal 

the  effect  of  '  alternating '  rather  than  superposed  whorls  or  cycles. 
resp  •  mode  of  arrangement  are  so  few.4  that 

the  diagonally   intersecting  curves   of  the 
undoubtedly  accepted  as  the  original  lines  of  the  "general 
constructions. 

III.  Comparison  of  such 'diagonal 

ior  'spiral')  construction   must  be  regarded  as 
syn.  ..hurled')  homologue  is  \ 

more  characteristic  of  parts  of  the  pi 
synthetic    :1  oral -shoot),  or  it  occurs  in  types  of 

has  been  ideologically  identified  with 
superposition  and  reduced  transpiration.    The 
of  asymmetry  is  thus  confirmed  as  being 
and  the  systems  of  alternating  whorls  as  special 
expressing,  by  the  retention  ot 
trated  '  plan,  their  derivative  nature.1 

1  R.PJM.U,  tec.  at,  p.  no. 

«  Bower  (1908),  '  Origin  of  Land-rV: 

*  •  Annals  of  **"*•"•    •*gSa'"feA^  •*  •»• 


1904),  p.  *jj- 

4  a.  a  few  isolated  examples  of  doafeM 
nhytic  orgmnisatioo ;  Cbiritfnlmt  (Scou) ; 
•  RJ.M.L.,  tec.  dt,p,  141. 


52        On  the  Interpretation  of  Phenomena  of  P/iyl lot  axis. 

IV.  Among    the   more    primitive   'spiral1    constructions,    the    ; 
numerical  preponderance  of  Fibonacci  ratios  shows  that  these  must  be 
taken   as  the   more  elementary,  and   most  probably   the  original   case. 
Exceptions  commonly  occur  as  simple  factorial  variants  (bijugate  systems, 
&c.),  and  the  few  '  anomalous  *  systems  show  simple  divergence  from  the 
^  ratio  of  i  :  1-618.     All  these  latter  again  tend  to  approach  this  r.uio  as 
the  systems  rise  higher  in  the  numerical  scale.1     The  extreme  divergence 
of  all  'spiral*   systems    lies   between  the   ratio    i:i    and    i :  2,   with   an 
optimum  at  137$° ;   and  all  anomalous  systems  tend  to  approach  nearer 
the  i  :  i  ratio  of  equality,  rather  than  the  i  :2  side  of  this  *  ideal  angle';2 
in    no  case  passing  beyond  it.      Thus,  taking  the  possible  combinations 
of  the  numerals  3  and  4 — out  of  3  :  3  (symmetrical),  3  : 4  (anomalous),  3  :  .5 
(Fibonacci),  3  :  6  (trijugate) — all  occur :  the  same  applies  to  4  :  4  (whorled), 
4  :  5  (anomalous),  4  : 6  (bijugate) ;   but  4  :  7  does  not  occur,  and  4 :  8  has 
not  been  recorded.     The  simplest  anomalous  cases  3  :  4  (cf.  Scdum),  and 
4:5   (cf.  Lycopodium),  of  the  type  m  :  M+I.  arc  those  found,  and  they 
are  generally  distributed.     The   fact   that   the    Fibonacci   relation  is  the 
commonest,   most   widely   distributed,  and    therefore   possibly   the    most 
primitive   type,  follows   from  purely  morphological    considerations,   quite 
apart  from  any  Ideological  explanation  of  its  assumed  advantage  in  the 
case  of  land -vegetation ;  as  in  the  older  views  of  Bonnet,  which  assumed 
optimum  advantage  for  transpiration,  or  the  later  improved  view  of  Wiesncr 
that  it  gives  the  optimum  angle  for  maximum  exposure  to  light  in  photo- 
synthesis.3   The  latter  standpoint  covers  and  amplifies  that  of  Bonnet ; 
but  it  does  not  follow  that  even  this  view  contains  the  whole  truth,  or 
is  even  near  it.     It  still  remains  to  explain  (i)  How  the  plant  ever  found 
the  angle,  (2)  The  original  mechanism  of  production ;  and  the  conclusion 
is  immediately  suggested  that,  even  for  purposes  of  photosynthesis,  the 
construction-system  has  been  '  adapted  ',  and  there  may  be  still  something 
behind. 

V.  But  having  got  thus  far.  one  can  look  back  and  see  the  vast  range 
of  secondary  phenomena  covered  by  '  compromises '  between  positions  of 
4  maximum    exposure'   and    'minimum   exposure'    (superposition),   which 
make  up  the  systems  of  the  leafy-shoots  of  land-plants  ;  as  also  the  remark- 
able fact  that  the  most  perfect  expressions  of  Fibonacci-relations,  and  again 
the  most  constant  in  occurrence,  are  always  to  be  found,  not  in  the  more 
perfectly  equipped  photosynthetic  shoot-systems,  but  in  constructions  in 
which  the  demand  for  photosynthetic  exposure  is  nil\  e.g.  in  the  scales 
of    Conifer   cones,   the    inflorescence-capitula    with    generally   suppressed 
'  bracts '  of  Composites,  and  the  stems  of  leafless  Cacti.     The  conclusion 
is  forced  upon  one  that  there  must  be  after  all  some  further  object  in  view, 
as  the  expression  of  some  still  more  fundamental  law  of  living  organism,  of 
which  leaf-arrangement  is  but  one  special  case.     The   only  satisfactory 
generalization  behind  the  utilization  of  the  Fibonacci  ratio  in  land-plants 
is  that  if  the  plant  for  some  reason  '  prefers '  or  is  bound  to  build  one  new 
member  at  a  time,  the  Fibonacci  angle  is  undoubtedly  the  one  to  approxi- 
mate.    But  there  is  no  apparent  reason  so  far  put  forward  as  to  why  the 
land-plant  should  be  so  handicapped.     Examples  of  whorled  systems,  and 
their  mechanism  for  the  production  of  members  simultaneously,  show  that 
this  can  be  done  with  equal  facility ;   yet  whorled  systems  appear  as  an 


1  R.P.M.L.,  loc.  cit.,  p.  72. 
-  loc.  cit.,  p.  197. 

3  Little  better  than  the  original  view  of  Leonardo  da  Vinci;  cf.  Cook  (1914)1 
4 The  Curves  of  Life',  p.  81. 


Pkylttic  Progression  in  Pkyllotaxii-mmktmitm* 

afterthought.     It  can  be  only  concluded  that  the  plant  is  some* 
•11  the  first  in  favour  of  members  imagod  one  by  one  ia  a 
sequence ;  and  the  suggestion  immediately  offer*  that  tali  may  be  in 
way  the  expression  of  the  inheritance  of  the  equipment  of  a  prec 
phase  and  the  solution  of  a  much  older  problem. 

VI.  Thcca^M:f>(theMo«^ametc^yU>whkh*timiUfre»ttJtU 
;>i«irently  as  the  effect  of  a 

more  probably,  since  the 
oblique,  only  associated 

g  link  with  still  lower  races  of 
nmcnt  of  the  sea.  in  which  all  the 
similarly  obtain  ;  though  no  horizontally 
demanding  exposure  to  vertical  light  are  ia  existent*,  or  nave  been  as  pal 
evolved.  The  seaweed  (as  cxpicssui  in  the  Focoid  illianm)  is  caeaJly 
under  the  necessity  of  building  one  lateral  member  at  a  time  <thm*h,  ia 

case  an  obvious  •  branch  '-ramulu*.  rather  than  a  highly  specBi 
•  leaf '.member),  and  the  name  consequences  of  Fibonacci  mfmimm  art 
be  observed  Whether  the  mechanism  of  production  is,  or  if  ant,  asjad 

the  nuclear  mcchanUm  of  apical-cell  MfinanlUlofX  If  dearly 
material.  The  system  is  undoubtedly  capable  of  sscoadary  adjastfl 
(as  in  the  bilateral  Haltdrys],  whatever  may  be  the  angle  ffvea  by  tar 
routing  nuclear  spindle  of  the  apical  cell  Finally  the  case  of  the  feeae* 
seaweed  is  extended  to  the  limiting  expression  of  the  nolotok  Fc 
in  which  li.'.ht-ertcct  is  completely  ruled  out  of  the  problem .  and  la 
of  an  entirely  different,  or  even  diametrically  opposed 
same  Fibonacci  result  follows,  even  with  recognizably 

VII.  The  last  case  gives  the  conclusion  needi 
for  building  one  member  at  a  time  (whatever  the  ' 
the  significant  factor,  to  which  all  others  are  « 

The  question  of  Fibonacci  orientation  thus  reduces  to  a  function  of  prisaay 
centric  growth  correlated  with  secondary  asymmetrical  increase  hi  a  ajsMam 
mass,  within  the  operation  of  surface-tension,  and  so  reduces  hi  the  Wail  la 
the  fundamental  problem  of  the  struggle  of  " 


in  terms  of  surface-exposure  to  the  medium  of  the  sea*  to 

thing  more  than    minimum  surface.1     If  in  such  a   pi 

organism,  outgrowth  in  one  direction  is  followed  by  a 

movement  in  the  next  position  for  balanced 

of  the  latter  implies  the  presentation  of  a 

Fibonacci  series,  as  an  inevitable  consequence. 

movement  out  the  Fibonacci-pattern,  as  a 

the  symmetrically  placed  growth -ex  tensions  of 

helicoid  spiral  of  other  Foramintfcra,  or  simple  *  bilateral* 

plane.    The  special  advantages  of  the  Fibonacci  rhythm  bea* 

maximum  compactness  of  the  resultant  soma,  and  <t)  its  capacity  for 

in<i< finite growA-fxtemsu*  o*  tke  $*me  terms. 

With  tliis  :<..:   i.imcntal  gencralizatioi 
on  which  to  build,  it  is  possible  to  begin  to 
such  organism  along  its  upward  path.     In  the 
multiscptatc  phytobcnthon  of  the  sea,  the  nccessi  _ 
surface  by  rtitnifitatunt,  follows  as  the  natural  rfipoasc  to  the 

moving  medium,  in  which  the  mala  axis  requires  to  be 
adjusted  to  resist  the  strain  of  wave-tension.    Such  | 
increases  the  somatic  form  along  similar  lines,  and 

1  BOL  Mess.  i. loc.au  a  it 


54        Ou  Hit  Interpretation  of  Plwiomena  of  Phyllota.\ 

Any  point  of  the  surface  may  '  throw  out '  a  new  branch,  exactly  as  any 
part  of  a  bcnthic  amoeba  may  throw  out  a  local  '  pseudopodium  '. 
amples  of  such  elementary  benthic  growth-forms  arc  common  to  the  pr< 
day  in  simpler  Phacophyccae  (cf.  Mesogloia\  and  are  also  abundant  am 
Floridcae,  traced   in  the  isolated  flagellate  progression  of  Ifytfrnrus,  and 
the  case  of  the  Sckisonctna- Diatom.      The  last  vestige  of  such  arc! 
irregular   ramification   may  be   traced    in   higher   land-vegetation   as   the 
•  adventitious  branch '  or  even  root.     But  with  the  progression  of  phyto- 
bcnthon  to  more  localized  and  apical  growth  in  the  main  axis,  ramification 
becomes  increasingly  restricted  to  the  distal  or  apical  region  of  the  shoot  ; 
the  adventitious  character  is  diminished  and  finally  suppressed,  as  the  older 
portions  in  attaining  an  adult-phase  lose  the  capacity  for  initiating  new  and 
young  growths.     As   again,  taking  well-known  examples  in   the  case  of 
higher  vegetation,   the   retention  of    pericyclic   ti  i   a   permanently 

juvenile  condition  renders  possible  the  production  of  adventitious  roots, 
and  wound-callus  may  regenerate  cither  new  stem  or  new  root-apices.  In 
all  such  cases  the  primary  irregularity  of  ramification  settles  down  to  the 
production  of  individual  ramuli  one  at  a  time,  in  acropetal  series ;  and 
the  necessity  for  equal-spacing,  as  a  balancing  of  the  symmetry  of  the 
shoot,  follows  as  naturally  as  the  retention  of  its  radial  organization  and 
cylindrical  form.  Still  more  perfectly  can  such  a  mechanism  be  established 
when  the  segmentation  of  the  apex  itself,  within  the  control  of  a  single 
dominant  apical  cell,  acquires  precision  in  the  centric  distribution  of  the 
growth-forces.  But  it  remains  abundantly  clear  that  the  possibilities  of 
Fibonacci  orientation  in  the  branch-ramuli  are  older  than  such  specialized 
apical  differentiation,  as  they  arc  far  older  than  the  differentiation  of  even 
leaf-laminae.  Fibonacci  symmetry  is,  in  fact,  one  of  the  most  archaic  of 
somatic  factors ;  it  is  difficult  to  trace  anything  phyletically  more  remote. 
This,  again,  undoubtedly  affords  the  clue  to  its  extreme  persistence,  even 
when  no  longer  an  integral  part  of  shoot-construction  ;  e.g.  the  retention  of 
a  complex  mechanism  for  the  distribution  of  groups  of  spines  on  a  leaflc-s 
Cactus^  or  the  predominant  retention  of  the  number  5  in  whorled  petaloid 
flowers.  While  the  case  of  the  Moss  now  appears  as  a  joint  association  of 
the  apical  mechanism  of  cell-differentiation  with  the  Fibonacci  orientation 
of  more  definite  leaf-laminae, — to  the  extent  that  the  latter  is  quite  as  pos- 
sibly the  causal  factor,  in  the  presentation  of  the  limiting  case  of  the  3-sided 
centric  apical  cell,  as  the  more  obvious  mechanism  of  the  cell  itself, — the 
obvious  dissociation  of  the  two  factors  in  the  Pteridophyta  only  amplifies 
the  story  of  the  Fucoid  ;  and  henceforward  there  is  no  doubt  whatever  that 
these  two  functions  of  a  growing-apex  are  entirely  independent  of  one 
another.  The  case  of  Eqnisctum  remains  to  make  this  perfectly  clear.1 
On  the  other  hand,  in  the  case  of  higher  plants,  it  is  evident  that  there  must 
be  some  new  apical  mechanism  of  control,  which  involves  the  working  out 
of  such  perfect  patterns.  So  long  as  the  numerical  expression  of  these 
ratios  remains  low,  the  appearance  of  building  one  member  at  a  time  works 
out  as  the  familiar  'genetic  spiraP-effect  of  the  adult  shoot-system.  Hut 
with  numerical  increase  of  the  ratios,  as  the  effect  of  a  diminished  diameter 
of  the  lateral  centres  in  comparison  with  the  diameter  of  the  axis,  many 
members  are  being  formed  practically  simultaneously  (e.g.  scores  in  the 
case  of  a  Composite  capitulum,  or  even  hundreds  in  the  classical  example 
of  the  great  Sunflower  heads,  working  in  terms  of  89: 144,  at  least  a  full 
contact-cycle  arise  simultaneously,  so  far  as  can  be  seen),  and  the  '  genetic 
spiral '  appears  as  a  useless  abstraction.  At  such  a  stage  it  now  becomes 


1  R.P.M.L.,  loc.  cit.,  p.  150. 


Pkyl.'i    I'rogrtsrion  in  l*kyllotajrit-mMd*mism,  55 


obvious  that  the  rnnrhinitm.  having  lost  its  initial  (actor  and 

begin  to  become  irregular  and  anomalou* .  and  hence  it  is  to  feral  can* 

•tractions  involving  small  reproductive  members  of  no 

that  the  system  begins  to  break  down ;  as  in  the  gym 

berry,  the  sporophvlls  of  Clrmttu.  or  the  androecfaMB  of  the  Poppy. 


last  become  familiar  examples  of  sheerly  irregular  ijsnmi  •iih  oatv 
vestigial  rhythmic  effects  ;  both  the  idea  of  one  member  at  a  time  and  that 
of  Fibonacci  symmetry  being  hopelessly  lost,  nothing  is  left  bet  a  vaftt 
acropetal  sequence.  Even  this  is  open  to  alteration,  as  to  the  example  of 
ndrocciura  of  the  Pacony.  Finally,  by  loss  of  all 


the  shoot-system    returns    to  an  almost   algal-tike   phase  of 


enations1,  as  postulated  in  conventional 

At  this  point  one  is  again  brought  up  against  the  (act  that  the  para 

only  constant  and  fundamental  feature  of  UK 
tions.    All  irregularities  in  the  system  reduce  to  variations  to  the 
constants ;  and,  bearing  in  mind  the  (act  that  the  isophylly  of  the  r 
as  expressed  in  quasi-circle  origin,  postulate!  an  orthogonally  atfBMated 
system,  one  is  driven  to  the  conclusion,  wkttkfr  ma  Ufa  u  *  •*.  that  the 


apex  presents  a  certain  capacity  for  numerical  choice  in  the 
expression  of  certain  intersecting  paths  of  equal  distribution  of  the  erowth- 
forces  which  may  be  included  within  the  convention  of  earfpiitfwltoi  The 
ingrained  habit  of  building  one  member  at  a  time.inhsiNd  fan  the  tiMtfhif 
seaweed,  remains  curiously  dominant  even  in 
grade  ;  yet  at  any  time  the  claims  of 


>lution  of  the  problem,  as  expressed 
simultaneously  with  mathematical  precision,  from  the 


onwards ;  though  only  in  the  most  advanced  types  of 
ductive  shoots  (decussate  and  distichous  types)  can  it 

all  general.    Only,  again,  in  the  non-photosynthctii 

the  sporophyll-rcgion  of  higher  flowers,  does  it  attain  aay 

degree  of  predominance,  and  in  a  manner  which  on  aay  ether  caosml  or 

merely  ideological  interpretation  remains  wholly  unsatisfactory  and  even 

unintelligible.     Thus  in  a  simple  flower,  as  a  Rose  or  a 

retention  of  a  quincuncial  calyx  in  the  floral  organisation 

obvious  relic  of  a  mode  of  growth  older  than  the  first 

foliage-leaf,  of  which  the  sepals  arc  but  the 

again  the  accurately  whorlcd  and  alternating 

Columbine  or  a  Geranium  could  have  * 

primary  and  ancient  equipment      It  it  the 

minute  and  apparently  trivial,  or  wholly 

tclligible,  details  of  the  organization  of 

remotest  epoch  of  the  progression  of  the 

such  morphological  investigations  of 

i  vision  of  the  distant  vistas  of  the 
correct  perspecti 

There  can  be  no  doubt  that  in  the 
plant-life  from  the  antecedent  phases  of  pi 


does  begin  to  obtain  a  glimpse  of  the  master  to 
fundamental  principles  of  somatic  organitaUon  1 
morphological  construction  which  is  destined  to 
tul  method  of  initiating  new  c 
of  the  soma.  to  be  ultimately 
as  'appendages'  of  special 


production,  in  terms 

,1  closely  identical  morphological 


56        On  the  Interpretation  of  Phenomena  of  Phy  Hot  axis. 

the  unicellular  soma,  the  mechanism  is  traced  to  the  benthic  alga,  ultimately 
with  multiscptatc  axis  and  a  segmenting  apical  cell,  to  still  more  massive 
growths  in  which  the  apical  cell  loses  its  domination,  to  be  replaced  by  more 
obscure  growth -processes  with  apical  control ;  the  latter  appearing  at  the 
apex  of  the  land-plant,  in  which  the  member-producing  function  of  the 
apical  cell  is  entirely  superseded,  though  the  construction  stili  in  all 

essentials  its  Fibonacci  symmetry  (Filicineae),  or  a  special  case  of  symmetry 
readily  derivative  from  these  relations  (Equist'tum\t  together  with  many 
.nts  and  decadent  stages  (Filicineae,  Lycopodineae).  The  initiation  of 
the  primary  ramuli  (now  distinguished  as  leaf-appendages)  follows  a  third 
method  of  production,  and  this  mechanism  remains  as  the  characteristic 
expression  of  all  higher  plant-forms.  One  may  not  yet  see  exactly  how  it 
is  done,  as  a  more  intimate  plasmic  or  even  '  molecular'  function,  and  the 
equipotential  theory  so  far  is  helpful  as  covering  all  the  facts  of  observation ; 
but  that  phyllotaxis-mechanism  has  passed  through  successive  phases  of 
evolutionary  progression,1  and  is  by  no  means  to  be  explained  by  subaerial 
botanists  as  a  condition  of  casual  adaptation  to  the  state  of  the  plant  as  now 
found  growing  on  the  land-surface,  much  less  to  be  lightly  interpreted  along 
teleological  lines  of  the  modern  world,2  appears  at  present  the  surest  foun- 
dation on  which  to  erect  hypotheses  of  the  evolution  of  what  is  termed 
'stem'  and  'leaf.  Academic  abstractions  of  'caulome'  and  'phyllomc' 
are  meaningless  expressions  in  view  of  the  broader  outlook  which  demands 
some  definite  information  as  to  why  a  plant  is  what  it  is,  in  terms  of  cells, 
tissues,  members,  and  space-form. 

Taking  the  general  progression  of  Fibonacci  phyllotaxis  as  the  expres- 
sion of  an  archaic  method  of  initiating  one  lateral  extension  of  the  soma  at 
a  time,  from  a  growth-centre  or  a  differentiated  growing-point, — a  process 
which  may  be  continued  indefinitely  with  optimum,  self-regulated,  balanced 
symmetry,  undoubtedly  on  the  whole  the  most  satisfactory  solution  of  the 
problem  of  indefinitely  continued  two-dimensional  extension — the  more 
fundamental  and  primary  relations  of  living  plasma,  established  once  for  all, 
even  in  the  plankton-phase,  may  remain  predominant,  with  little  or  no 
change,  throughout  all  future  phases  of  progression,  as  if  their  value  might 
not  be  questioned.  This  has  been  seen  to  apply  to  the  photosynthetic 


1  Centric  symmetry  may  be  said  to  characterize  Coelenterata,  as  bilateral  and 
dorsiventral  symmetry  prevails  in  all  phyla  of  originally  creeping  and  benthic  Metazoa  ; 
centric  asymmetry,  inevitably  involving  Fibonacci-relations,  dominates  the  plant- 
kingdom  ;  other  phases  of  symmetry  (whorled  and  dorsiventral),  being  of  secondary 
significance  only.  Spiral  effects  are  equally  secondary  or  subjective,  as  Fibonacci 
symmetry  is  seen  to  be  the  expression  of  an  oscillatory  balancing  effect  in  two 
dimensions.  All  phases  of  somatic  symmetry  date  to  the  earliest  benthic  forms  in 
which  elaborate  somatic  organization  was  first  evolved  ;  the  main  groups  of  organisms 
diverging  along  their  special  lines,  the  more  widely  as  subaerial  transmigrants. 

9  Analogies  are  not  wanting  in  other  departments  of  biology;  for  example, 
a  man's  nose,  with  distinctly  heritable  minor  details,  is  derived  from  the  pointed  end 
of  the  body  of  a  benthic  fish ;  the  latter  expresses  the  pointed  end  of  a  flagellate, 
overhanging  the  primary  oral  aperture  (cytostome),  in  turn  the  consequence  of  a 
phase  of  elementary  polarity  beyond  the  original  surface-tension  sphere  of  aqueous 
plasma,  and  so  far  tracing  back  to  phenomena  associated  with  surface-tension.  Yet 
few  would  suggest  that  the  nose  is  modelled  in  the  human  embryo,  at  the  present 
time,  solely  as  a  result  of  surface-tension.  As  the  organism  becomes  more  complex, 
so  the  mechanism  producing  it  may  be  elaborated  beyond  recognition,  or  new 
mechanism  may  replace  the  old ;  such  mechanism  being  not  only  individual  but 
racial ;  i.  e.  representing  inherited  response  to  conditions  possibly  no  longer  effective. 


n  c« 


mechanism  of  the  chloropUfU  of  early  autotrophk 

little  changed  in  the  vegetation  of  the  Und  U>3ay;  tt  evotofe 

nucleated  cell  as  the  pUniaon-soroa,  still  the  unit  of  all 


plant  and  animal ;    as  also  to  the 
septate  axis  of  incipient  phytobcnthon 
cation  and  ultimate  delimitation  of 
il  the  construction-factors  of  the 

the  first  problems  of  benthic  exigence,  the  more  ingrained  do  they  i 
in  the  structural  mechanism  of  the  race  lor  all  fir 
difficult  to  eradicate.     Few  more  striking  illustration*  in 
are  available  than  the  retention  of 
a  group  as  the  Cactaceae,  where 


uppres*<  majority  of  phyla  of 

happened  to  follow  the  vr^-  rganixation  of  thit 

scries,  we  .should  have  heard  little  about  spiral  phvllouxit  being  threats* 
tion  of  the  problem  of  optimum  di  leave*  to  feddw  I%M 

The  occurrence  of  Fibonacci   ratio*  in   plant  organisation,  originally  the 
expression  of  balanced  symmetry  within  certain  limitation*,  this*  penssU  to 
the  present  day  throughout  the  great  range  of  modern  lind  flora ;  and  on 
the  whole  proves  equally  satisfactory  as  applied  to  the  problem*  of  tcatiftl 
on  and  light-utilization.    Though  by  no  mean*  the  only  MOUOSI 
possible,  it  happens  to  be  the  one  given  by  inheritance  from  \ 
c  phases,  and  hence  remains  largely  unaffected  in  the 

;  compensatory  corrections  may  be  added  m  term*  of 
secondary  growth- phenomena.  Only  in  the  general  case  of 
complex  floral  organization,  can  an  attainment  of  secondary  whmled  sym- 
metry be  said  to  be  at  all  characteristic  ;  and  in  this  case,  again,  it  i*  practi- 
cally confined  to  the  sporophylls  as  reproductive  members.  Hence  to 
higher  petaloid  flowers  the  change  is  associated  with  the  tTtiHntfin  of 

:oll.t  '-members  (Dicots.).  or  involve*  member*  of  the 
(Monocots.) ;  while  the  calyx  (perianth),  itself  vestigial  retain 
dcrful  conservatism,  otherwise  wholly  unintelligible,  indication* 
nacci  origin  ;  even  in  many  cases  (whole  families)  in  which  the 
of  the  vegetative  shoot  may  have  been  similarly  changed  to  syi 
construction  (eg.  a  :  a.  or  decussate),  as  in  Dentamerousflowtfs  of 
Gcntianaceae,  Logan iaccac,  Apocynaccac,  &c. 

The  futility  ot  attempting  to  reach  a  final  solution  of 
by  mere  observation  of  the  mutual  relations  of  the  lateral 
appendages  of  transmigrant  Land- Flora,  may  be  now  admitted.  All 
structural  relations  trace  back  to  the  sea.    The 
of  stem  and  root,  leaf  and  bran, 


IH  and  *po*» 

form,  is  to  be  sought  far  behind  the  comparatively  modem  and  ws* 


secondary  subaerial  environment  in  which  we  find 
familiar  vegetation  of  the  land.  That  early  botanical  writer*  lived  to  %M£ 
ancc  of  this  fact  excuses  their  many  limitations ;  at  the  present  day  a  *Mch 
broader  perception  is  possible,  and  with  the  upentog  up  of  the  mdetee 

is  of  life  ,  >n  this  world,  as  a  cooling  planet,  much  of  the  older  and  i 
academic  outlook  requires  re-orientation.    To  nuny  botanist*  this  *ort 
general  coiu  n.u   still  appear  fantastic,  at  savoarin*  of 

'hilosophy  ' ;  but  it  may  be  pointed  out  that  the  '  Nature*  ef  the 


y  be  undcrstocxl  the  progression  of  We  onthiiworfc 
real  phenomenon,  still  demanding  a  philosophy  forte  toterpfetaoosiai 

scntation,  as  well  as  encouraging  the  met 


\  i 


LIST    OF    FIGURES 

FIG.  I.  Geometrical  construction  for  uniform  centric  growth-expansion,  show- 
ing method  of  obtaining  orthogonally  intersecting  pairs  of  log-spirals  for  any  required 
ratio,  symmetrical  or  asymmetrical,  to  be  used  as  curve-rules  for  drawing  any  required 
construction  as  a  standard  of  reference. 

FIG.  II.  Centric  spiral  construction  (8:13)  in  terms  of  quasi-square*  wiih 
inscribed  quasi -circles. 

FIG.  III.  Transformation  of  system  (5:8)  to  eccentric  homologue  (zygo- 
morphic),  orientated  in  plane  of  No.  2. 

Fio.  IV.      Centric  symmetry  (5 :  5),  system  of  whorled  pentamery. 

FIG.  V.  Transformation  of  (5 : 5)  to  eccentric  homologue  (zygomorphic 
pentamery). 

FIG.  VI.     Pinus  Pinea,  transverse  section  of  apex  of  young  seedling,  6in.  high, 
n  (5:8),  approximating  quasi-square  construction   in  terms  of   needle-leaves 
under  mutual  contact-pressures. 

FIG.  VII.  Euphorbia  Wul/cnii,  apex  of  strong  axis  (8:  13),  with  progressive 
dorsiventrality  of  members,  and  correction-effects  of  sliding-growth. 

FIG.  VIII.  Dipsacus  fullonum,  theoretical  construction  for  inflorescence-scheme 
of  involucral  members  and  florets  (16:26). 

FIG.  IX.  Set  of  5  quasi-circles  of  the  systems  (3 :  5),  (2  13),  (2:2),  (i  :  2), 
(i  :  i),  arranged  for  convenience  in  diminishing  series,  i,  2,  3,  4,  5,  respectively,  along 
the  plane  of  median  bilaterality  X  7. 

Cr  Ct,  C9.  C4,  C#  the  centres  of  construction,  and  0,,  04.  O^  the  origins 
for  respective  curves. 

A  circle  AB  with  centre  C,  has  been  drawn  in  contact  with  the  (3  :  5)  curve 
for  purposes  of  comparison. 

FIG.  X.  Araucaria  exctlsa,  apices  of  lateral  axes  (7:  n),  (5:8),  (3:5),  as 
expressions  of  bulk-ratio. 

FIG.  XI.  Araucaria  exceha,  lateral  axis  cut  obliquely  to  show  origin  and  orienta- 
tion of  laterals  of  next  degree. 

FIG.  XII.  Scmpervivum  cakara/um,  transverse  section  of  apex,  system  (3  : 5), 
for  measurement. 

FIG.  XIII.  Quinqutloculina  scminulum,  transverse  section,  system  (2:3),  from 
a  specimen  in  calcite. 

FIG.  XIV.     Cobata  scandens,  apex  of  vegetative  shoot,  transverse  section,  (3:5^ 

FIG.  XV.  Ltpidoslrobus  sp.,  transverse  section  of  axis  showing  trace-bundles  in 
system  (8: 13). 

XVI.     Polytrichum  commune,  transverse  section  of  leafy  stem,   showing 
pattern  of  trace-bundles  in  the  cortex. 

FIG.  XVII.     Cystostira  ericoidts,  apices  of  shoots  as  seen  from  above. 

FIG.  XVIII.  Retardation-effects  in  distichous  (i  +  i)  system:  A  quasi-circle 
in  relation  to  successive  members  of  the  series,  Cl  as  centre  of  construction  of  the 
lateral  primordium  :  Bt  C,  derivative  curves  with  radial  retardation  only :  D,  E, 
retarded  quasi-square  derivatives. 


•  „•:,.,  ,  »: 


BMlbod 

ratio,  symmetric*)  or  Mjmmttricml.  to  be 


FIG.  II.     Centric  spiral  construction  (8  :  13)  in  terms  of  quasi-squares 
\\ith  inscribed  quasi-circles. 


I.    Tramfennarion  of  17*00(5  :  8)io«cc««tffc 
orirnuird  in  plane  of  Ha  f . 


Fia.  IV.     Centric  symmetry  (5  :  5),  system  of  whorled  pentamery 


of  (s  :  s)  10  KOtMfc 


FIG.  VI.     Pinus  Pinta,  transverse  section  of  apex  of  young  seedling,  (>  in.  high, 
i.  ;q»proximating  quasi-square  construction  in  terms  of  needle-leaves 
uiul«-r  mutual  contact-pressures. 


\  II      /.W/Atrto  IM/nii.  apex  of 
dorei  venmdily  of  mcmben.  and 


FIG.  VIII.     Dipsacus /ullonum,  theoretical  construction  for  inflorescence- 
scheme  of  involucral  members  and  florets  (16  :  26). 


uati-circles  of  the  systems  (j  :  5).  (' 

(i  :  i).  arrangcil  for  convenu  hmmiahmg  •erics,   • 

along  (he  plane  of  median  bihueraJit) 

C,,  C4,  Ct,  the  icnirrs  of  cofMtfUCtion.  and  Ot.  O,,  CJr  • 

cs. 

A  circK  AB  \\  nh  centre  C,  has  been  drawn  in  contact  wfch  nVr  (j 
for  pur|>oses  of  comparison. 


Fi',.  X.     Jr.///'  iiria  cxcclsa,  apices  of  lateral  axes  (7  :  1 1),  (5  :  8),  (3  :  5), 
as  expressions  of  bulk-ratio. 


V  K-  V 

*  vv 


.  1  ucral  axis  cut  obbqoeljr  lo 
oricnut ion  of  laterals  of  next  degree. 


FIG.  MI.     &Mj  mUaraiutn,  iranb\<  n  ol  aj  <  (3  :  -,),  for 

measurement. 


X 1 1 1.     Quin<ju<l<Kulina  ttminulum,  iMnivcrc  section.  Sfttein  (i  :  3).  from  • 


'o 


~ 


I 


FIG.  XV11I.  Retardation-effects  in  distichous  (i-M)  system:  A  qiusi-circle  in 
relation  to  successive  members  of  the  series,  Cl  as  centre  of  construction  of  the  lateral 
primordium :  B,  C,  derivative  curves  with  radial  retardation  only :  D,  Et  retarded  quasi- 
square  derivatives. 


OXFORD  \\ICAL    MEMOIRS 


i.    THE  BUILDING  OF  AN  AUTOTROPHIC   FLAGELLATE,  by  A.    II. 
CHURCH.     1919.     1 


2.    GOSSYP1UM  IN  PRELINNAEAN  LITERATURE,  l,y  li 

1      ,;>..•.  figs.  25. 


3.  THALASSIOPHYTA  AND  THE  SUBAERIAL  TRANSMIGRATION, 

by  A.  li  CH.     1919.     Pp.  95.  35.6^. 

4.  ELEMENTARY  NOTES  ON  STRUCTURAL  BOTANY,  h\  A.  II. 

;.     1919.     12  Lectun  Pp.  27. 

5.  ELEMENTARY  NOTES  ON  THE  REPRODUCTION  OF  ANGIO- 

SPERMS,  by  A.   H.  CHURCH.     1919.     10  Lccturt'-.sclucli 

Pp.  25. 

6.  ON  THE  INTERPRETATION  OF  PHENOMENA  OF  PHYLLOTAXIS, 

by  A.  H.  CHTRCH.     1920.     Pp.  58,   18  figs. 


Printed  in  England  at  the  Oxford  University  Press 


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